Properties

Label 24-483e12-1.1-c1e12-0-1
Degree $24$
Conductor $1.612\times 10^{32}$
Sign $1$
Analytic cond. $1.08315\times 10^{7}$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·3-s + 5·4-s − 3·5-s + 6·6-s − 2·7-s + 4·8-s + 15·9-s − 3·10-s + 14·11-s + 30·12-s − 2·14-s − 18·15-s + 15·16-s − 15·17-s + 15·18-s + 19-s − 15·20-s − 12·21-s + 14·22-s + 6·23-s + 24·24-s + 15·25-s + 14·27-s − 10·28-s − 12·29-s − 18·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 3.46·3-s + 5/2·4-s − 1.34·5-s + 2.44·6-s − 0.755·7-s + 1.41·8-s + 5·9-s − 0.948·10-s + 4.22·11-s + 8.66·12-s − 0.534·14-s − 4.64·15-s + 15/4·16-s − 3.63·17-s + 3.53·18-s + 0.229·19-s − 3.35·20-s − 2.61·21-s + 2.98·22-s + 1.25·23-s + 4.89·24-s + 3·25-s + 2.69·27-s − 1.88·28-s − 2.22·29-s − 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12} \cdot 23^{12}\)
Sign: $1$
Analytic conductor: \(1.08315\times 10^{7}\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{483} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.46491028\)
\(L(\frac12)\) \(\approx\) \(22.46491028\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T + T^{2} )^{6} \)
7 \( ( 1 + T + 2 T^{2} - 23 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( ( 1 - T + T^{2} )^{6} \)
good2 \( 1 - T - p^{2} T^{2} + 5 T^{3} + p^{2} T^{4} - 7 T^{5} + 7 T^{6} - 3 p T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} - p^{5} T^{10} - 7 p^{2} T^{11} + 31 p^{2} T^{12} - 7 p^{3} T^{13} - p^{7} T^{14} + p^{8} T^{15} - 3 p^{6} T^{16} - 3 p^{6} T^{17} + 7 p^{6} T^{18} - 7 p^{7} T^{19} + p^{10} T^{20} + 5 p^{9} T^{21} - p^{12} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
5 \( 1 + 3 T - 6 T^{2} - 7 p T^{3} - 22 T^{4} + 133 T^{5} + 202 T^{6} - 207 T^{7} - 436 T^{8} + 687 p T^{9} + 10508 T^{10} - 2841 p T^{11} - 91434 T^{12} - 2841 p^{2} T^{13} + 10508 p^{2} T^{14} + 687 p^{4} T^{15} - 436 p^{4} T^{16} - 207 p^{5} T^{17} + 202 p^{6} T^{18} + 133 p^{7} T^{19} - 22 p^{8} T^{20} - 7 p^{10} T^{21} - 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 14 T + 78 T^{2} - 160 T^{3} - 303 T^{4} + 1876 T^{5} + 1018 T^{6} - 23850 T^{7} + 41202 T^{8} + 101126 T^{9} - 276378 T^{10} - 1801636 T^{11} + 11253009 T^{12} - 1801636 p T^{13} - 276378 p^{2} T^{14} + 101126 p^{3} T^{15} + 41202 p^{4} T^{16} - 23850 p^{5} T^{17} + 1018 p^{6} T^{18} + 1876 p^{7} T^{19} - 303 p^{8} T^{20} - 160 p^{9} T^{21} + 78 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \)
13 \( ( 1 + 47 T^{2} + 4 p T^{3} + 970 T^{4} + 2024 T^{5} + 13727 T^{6} + 2024 p T^{7} + 970 p^{2} T^{8} + 4 p^{4} T^{9} + 47 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 15 T + 62 T^{2} - 151 T^{3} - 1286 T^{4} + 4457 T^{5} + 52922 T^{6} + 164193 T^{7} + 160264 T^{8} - 1686289 T^{9} - 8188072 T^{10} + 47329399 T^{11} + 456657486 T^{12} + 47329399 p T^{13} - 8188072 p^{2} T^{14} - 1686289 p^{3} T^{15} + 160264 p^{4} T^{16} + 164193 p^{5} T^{17} + 52922 p^{6} T^{18} + 4457 p^{7} T^{19} - 1286 p^{8} T^{20} - 151 p^{9} T^{21} + 62 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 - T - 48 T^{2} - 173 T^{3} + 1914 T^{4} + 9399 T^{5} - 19398 T^{6} - 380937 T^{7} - 383790 T^{8} + 6839983 T^{9} + 35401704 T^{10} - 3715627 p T^{11} - 802999834 T^{12} - 3715627 p^{2} T^{13} + 35401704 p^{2} T^{14} + 6839983 p^{3} T^{15} - 383790 p^{4} T^{16} - 380937 p^{5} T^{17} - 19398 p^{6} T^{18} + 9399 p^{7} T^{19} + 1914 p^{8} T^{20} - 173 p^{9} T^{21} - 48 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
29 \( ( 1 + 6 T + 108 T^{2} + 456 T^{3} + 5671 T^{4} + 18570 T^{5} + 191272 T^{6} + 18570 p T^{7} + 5671 p^{2} T^{8} + 456 p^{3} T^{9} + 108 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( 1 - 11 T - 66 T^{2} + 1067 T^{3} + 3054 T^{4} - 60391 T^{5} - 104796 T^{6} + 1939181 T^{7} + 6838282 T^{8} - 43414441 T^{9} - 361930790 T^{10} + 385258665 T^{11} + 14422607686 T^{12} + 385258665 p T^{13} - 361930790 p^{2} T^{14} - 43414441 p^{3} T^{15} + 6838282 p^{4} T^{16} + 1939181 p^{5} T^{17} - 104796 p^{6} T^{18} - 60391 p^{7} T^{19} + 3054 p^{8} T^{20} + 1067 p^{9} T^{21} - 66 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 5 T - 158 T^{2} + 713 T^{3} + 14386 T^{4} - 55525 T^{5} - 946672 T^{6} + 2915345 T^{7} + 49649826 T^{8} - 99976669 T^{9} - 2213777210 T^{10} + 1504098913 T^{11} + 87152410010 T^{12} + 1504098913 p T^{13} - 2213777210 p^{2} T^{14} - 99976669 p^{3} T^{15} + 49649826 p^{4} T^{16} + 2915345 p^{5} T^{17} - 946672 p^{6} T^{18} - 55525 p^{7} T^{19} + 14386 p^{8} T^{20} + 713 p^{9} T^{21} - 158 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
41 \( ( 1 - 18 T + 220 T^{2} - 1492 T^{3} + 8047 T^{4} - 22018 T^{5} + 110984 T^{6} - 22018 p T^{7} + 8047 p^{2} T^{8} - 1492 p^{3} T^{9} + 220 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 37 T + 761 T^{2} + 10914 T^{3} + 120293 T^{4} + 1061333 T^{5} + 7670238 T^{6} + 1061333 p T^{7} + 120293 p^{2} T^{8} + 10914 p^{3} T^{9} + 761 p^{4} T^{10} + 37 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 + 3 T - 160 T^{2} - 545 T^{3} + 13032 T^{4} + 46987 T^{5} - 671180 T^{6} - 2522393 T^{7} + 23884216 T^{8} + 90832027 T^{9} - 567500848 T^{10} - 1549369953 T^{11} + 14777331494 T^{12} - 1549369953 p T^{13} - 567500848 p^{2} T^{14} + 90832027 p^{3} T^{15} + 23884216 p^{4} T^{16} - 2522393 p^{5} T^{17} - 671180 p^{6} T^{18} + 46987 p^{7} T^{19} + 13032 p^{8} T^{20} - 545 p^{9} T^{21} - 160 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 15 T - 2 T^{2} - 369 T^{3} + 16458 T^{4} - 12903 T^{5} - 247604 T^{6} - 9701301 T^{7} + 539274 p T^{8} + 240277605 T^{9} + 3972192578 T^{10} - 22545014781 T^{11} - 108957509502 T^{12} - 22545014781 p T^{13} + 3972192578 p^{2} T^{14} + 240277605 p^{3} T^{15} + 539274 p^{5} T^{16} - 9701301 p^{5} T^{17} - 247604 p^{6} T^{18} - 12903 p^{7} T^{19} + 16458 p^{8} T^{20} - 369 p^{9} T^{21} - 2 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 - 2 T - 148 T^{2} - 312 T^{3} + 9833 T^{4} + 66474 T^{5} - 306364 T^{6} - 4757162 T^{7} - 1128486 T^{8} + 182811708 T^{9} + 1173811500 T^{10} - 2979365382 T^{11} - 110759718655 T^{12} - 2979365382 p T^{13} + 1173811500 p^{2} T^{14} + 182811708 p^{3} T^{15} - 1128486 p^{4} T^{16} - 4757162 p^{5} T^{17} - 306364 p^{6} T^{18} + 66474 p^{7} T^{19} + 9833 p^{8} T^{20} - 312 p^{9} T^{21} - 148 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 12 T - 58 T^{2} + 608 T^{3} + 6701 T^{4} - 30184 T^{5} - 202622 T^{6} + 276196 T^{7} - 11823878 T^{8} - 3623732 T^{9} + 2628991582 T^{10} - 1783183848 T^{11} - 182243772747 T^{12} - 1783183848 p T^{13} + 2628991582 p^{2} T^{14} - 3623732 p^{3} T^{15} - 11823878 p^{4} T^{16} + 276196 p^{5} T^{17} - 202622 p^{6} T^{18} - 30184 p^{7} T^{19} + 6701 p^{8} T^{20} + 608 p^{9} T^{21} - 58 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 - 10 T - 227 T^{2} + 1662 T^{3} + 34495 T^{4} - 140620 T^{5} - 4235652 T^{6} + 11241432 T^{7} + 400571277 T^{8} - 622340118 T^{9} - 33614705065 T^{10} + 13395804686 T^{11} + 2500028261238 T^{12} + 13395804686 p T^{13} - 33614705065 p^{2} T^{14} - 622340118 p^{3} T^{15} + 400571277 p^{4} T^{16} + 11241432 p^{5} T^{17} - 4235652 p^{6} T^{18} - 140620 p^{7} T^{19} + 34495 p^{8} T^{20} + 1662 p^{9} T^{21} - 227 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \)
71 \( ( 1 + 21 T + 503 T^{2} + 6682 T^{3} + 92121 T^{4} + 885629 T^{5} + 8722550 T^{6} + 885629 p T^{7} + 92121 p^{2} T^{8} + 6682 p^{3} T^{9} + 503 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 - 8 T - 159 T^{2} + 928 T^{3} + 10583 T^{4} - 7476 T^{5} - 558232 T^{6} - 197136 T^{7} + 43138593 T^{8} - 169024240 T^{9} - 7243328185 T^{10} + 6247058492 T^{11} + 800591978606 T^{12} + 6247058492 p T^{13} - 7243328185 p^{2} T^{14} - 169024240 p^{3} T^{15} + 43138593 p^{4} T^{16} - 197136 p^{5} T^{17} - 558232 p^{6} T^{18} - 7476 p^{7} T^{19} + 10583 p^{8} T^{20} + 928 p^{9} T^{21} - 159 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 17 T + 114 T^{2} - 17 p T^{3} + 15048 T^{4} + 3011 T^{5} - 238082 T^{6} - 7355881 T^{7} + 11653700 T^{8} + 275245233 T^{9} + 5236517222 T^{10} - 15492392133 T^{11} - 465761345450 T^{12} - 15492392133 p T^{13} + 5236517222 p^{2} T^{14} + 275245233 p^{3} T^{15} + 11653700 p^{4} T^{16} - 7355881 p^{5} T^{17} - 238082 p^{6} T^{18} + 3011 p^{7} T^{19} + 15048 p^{8} T^{20} - 17 p^{10} T^{21} + 114 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \)
83 \( ( 1 - 12 T + 418 T^{2} - 3758 T^{3} + 75163 T^{4} - 531066 T^{5} + 7859204 T^{6} - 531066 p T^{7} + 75163 p^{2} T^{8} - 3758 p^{3} T^{9} + 418 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
89 \( 1 + 18 T - 182 T^{2} - 5540 T^{3} + 16597 T^{4} + 1076934 T^{5} + 1890990 T^{6} - 134918690 T^{7} - 753734574 T^{8} + 11151303568 T^{9} + 125289577770 T^{10} - 404240701390 T^{11} - 13303951500731 T^{12} - 404240701390 p T^{13} + 125289577770 p^{2} T^{14} + 11151303568 p^{3} T^{15} - 753734574 p^{4} T^{16} - 134918690 p^{5} T^{17} + 1890990 p^{6} T^{18} + 1076934 p^{7} T^{19} + 16597 p^{8} T^{20} - 5540 p^{9} T^{21} - 182 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
97 \( ( 1 + 2 T + 354 T^{2} + 290 T^{3} + 58303 T^{4} - 14692 T^{5} + 6475708 T^{6} - 14692 p T^{7} + 58303 p^{2} T^{8} + 290 p^{3} T^{9} + 354 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.58743315325005279377968649464, −3.51072135777687319842508961262, −3.40820714743758520377596905568, −3.29894816254258549213863630150, −3.27034053515498270791376095632, −3.16416138126449895106517805066, −2.94169066048299724731576268157, −2.89483138249863713581717782760, −2.89414889421412008751985900622, −2.83401178898302317099528592050, −2.68057883470875350815521270895, −2.43248964052498885208705690508, −2.27501025536518949375066765892, −2.21720083322135796933028674679, −2.02349503920634885577049846436, −2.00571890132037533528108215260, −1.90468433398070680125357429628, −1.88196913396553043880581768061, −1.77618636358521110054097541609, −1.36912929656625307713918935196, −1.28530776677899387597840307719, −1.04268003487256538182380933392, −0.988473831032917368374634401854, −0.814090694750727083414636215371, −0.19246361678495828499050822838, 0.19246361678495828499050822838, 0.814090694750727083414636215371, 0.988473831032917368374634401854, 1.04268003487256538182380933392, 1.28530776677899387597840307719, 1.36912929656625307713918935196, 1.77618636358521110054097541609, 1.88196913396553043880581768061, 1.90468433398070680125357429628, 2.00571890132037533528108215260, 2.02349503920634885577049846436, 2.21720083322135796933028674679, 2.27501025536518949375066765892, 2.43248964052498885208705690508, 2.68057883470875350815521270895, 2.83401178898302317099528592050, 2.89414889421412008751985900622, 2.89483138249863713581717782760, 2.94169066048299724731576268157, 3.16416138126449895106517805066, 3.27034053515498270791376095632, 3.29894816254258549213863630150, 3.40820714743758520377596905568, 3.51072135777687319842508961262, 3.58743315325005279377968649464

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.