Properties

Degree 24
Conductor $ 2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s + 2·3-s + 78·4-s − 6·5-s − 24·6-s + 4·7-s − 364·8-s + 72·10-s − 7·11-s + 156·12-s − 2·13-s − 48·14-s − 12·15-s + 1.36e3·16-s + 17-s − 2·19-s − 468·20-s + 8·21-s + 84·22-s − 9·23-s − 728·24-s + 15·25-s + 24·26-s + 27-s + 312·28-s + 3·29-s + 144·30-s + ⋯
L(s)  = 1  − 8.48·2-s + 1.15·3-s + 39·4-s − 2.68·5-s − 9.79·6-s + 1.51·7-s − 128.·8-s + 22.7·10-s − 2.11·11-s + 45.0·12-s − 0.554·13-s − 12.8·14-s − 3.09·15-s + 341.·16-s + 0.242·17-s − 0.458·19-s − 104.·20-s + 1.74·21-s + 17.9·22-s − 1.87·23-s − 148.·24-s + 3·25-s + 4.70·26-s + 0.192·27-s + 58.9·28-s + 0.557·29-s + 26.2·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{630} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $0.0560984$
$L(\frac12)$  $\approx$  $0.0560984$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{12} \)
3 \( 1 - 2 T + 4 T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} - p^{5} T^{9} + 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + T + T^{2} )^{6} \)
7 \( ( 1 - 2 T - 4 T^{2} + 31 T^{3} - 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good11 \( 1 + 7 T - 9 T^{2} - 100 T^{3} + 262 T^{4} + 640 T^{5} - 8115 T^{6} - 15798 T^{7} + 74164 T^{8} + 669 p^{2} T^{9} - 972110 T^{10} + 548862 T^{11} + 17010145 T^{12} + 548862 p T^{13} - 972110 p^{2} T^{14} + 669 p^{5} T^{15} + 74164 p^{4} T^{16} - 15798 p^{5} T^{17} - 8115 p^{6} T^{18} + 640 p^{7} T^{19} + 262 p^{8} T^{20} - 100 p^{9} T^{21} - 9 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 + 2 T - 44 T^{2} - 132 T^{3} + 977 T^{4} + 3781 T^{5} - 11684 T^{6} - 65747 T^{7} + 50792 T^{8} + 714313 T^{9} + 868188 T^{10} - 3554073 T^{11} - 19510287 T^{12} - 3554073 p T^{13} + 868188 p^{2} T^{14} + 714313 p^{3} T^{15} + 50792 p^{4} T^{16} - 65747 p^{5} T^{17} - 11684 p^{6} T^{18} + 3781 p^{7} T^{19} + 977 p^{8} T^{20} - 132 p^{9} T^{21} - 44 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - T - 30 T^{2} + 115 T^{3} - 5 T^{4} - 2674 T^{5} + 5895 T^{6} + 18453 T^{7} - 1072 p T^{8} - 437961 T^{9} + 1559167 T^{10} + 7061088 T^{11} - 61824431 T^{12} + 7061088 p T^{13} + 1559167 p^{2} T^{14} - 437961 p^{3} T^{15} - 1072 p^{5} T^{16} + 18453 p^{5} T^{17} + 5895 p^{6} T^{18} - 2674 p^{7} T^{19} - 5 p^{8} T^{20} + 115 p^{9} T^{21} - 30 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 2 T - 80 T^{2} - 180 T^{3} + 3449 T^{4} + 7837 T^{5} - 102176 T^{6} - 211409 T^{7} + 2371040 T^{8} + 3579469 T^{9} - 47864088 T^{10} - 27533469 T^{11} + 916901667 T^{12} - 27533469 p T^{13} - 47864088 p^{2} T^{14} + 3579469 p^{3} T^{15} + 2371040 p^{4} T^{16} - 211409 p^{5} T^{17} - 102176 p^{6} T^{18} + 7837 p^{7} T^{19} + 3449 p^{8} T^{20} - 180 p^{9} T^{21} - 80 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 9 T - 48 T^{2} - 477 T^{3} + 2766 T^{4} + 20637 T^{5} - 3617 p T^{6} - 525717 T^{7} + 1781613 T^{8} + 8690382 T^{9} - 29166372 T^{10} - 91574172 T^{11} + 316597101 T^{12} - 91574172 p T^{13} - 29166372 p^{2} T^{14} + 8690382 p^{3} T^{15} + 1781613 p^{4} T^{16} - 525717 p^{5} T^{17} - 3617 p^{7} T^{18} + 20637 p^{7} T^{19} + 2766 p^{8} T^{20} - 477 p^{9} T^{21} - 48 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 3 T - 3 p T^{2} - 372 T^{3} + 6477 T^{4} + 37248 T^{5} - 116788 T^{6} - 2654523 T^{7} - 2866536 T^{8} + 76700583 T^{9} + 16545069 p T^{10} - 1248956703 T^{11} - 16840293807 T^{12} - 1248956703 p T^{13} + 16545069 p^{3} T^{14} + 76700583 p^{3} T^{15} - 2866536 p^{4} T^{16} - 2654523 p^{5} T^{17} - 116788 p^{6} T^{18} + 37248 p^{7} T^{19} + 6477 p^{8} T^{20} - 372 p^{9} T^{21} - 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
31 \( ( 1 - 9 T + 150 T^{2} - 1087 T^{3} + 9609 T^{4} - 58170 T^{5} + 367149 T^{6} - 58170 p T^{7} + 9609 p^{2} T^{8} - 1087 p^{3} T^{9} + 150 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 - 6 T - 3 p T^{2} + 802 T^{3} + 5847 T^{4} - 49353 T^{5} - 218618 T^{6} + 1971477 T^{7} + 7114131 T^{8} - 58413395 T^{9} - 181224618 T^{10} + 875260311 T^{11} + 4536500245 T^{12} + 875260311 p T^{13} - 181224618 p^{2} T^{14} - 58413395 p^{3} T^{15} + 7114131 p^{4} T^{16} + 1971477 p^{5} T^{17} - 218618 p^{6} T^{18} - 49353 p^{7} T^{19} + 5847 p^{8} T^{20} + 802 p^{9} T^{21} - 3 p^{11} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
41 \( ( 1 - 2 T + 118 T^{2} - 443 T^{3} + 5993 T^{4} - 38003 T^{5} + 232561 T^{6} - 38003 p T^{7} + 5993 p^{2} T^{8} - 443 p^{3} T^{9} + 118 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )( 1 + 13 T - 5 T^{2} - 1007 T^{3} - 5128 T^{4} + 21646 T^{5} + 337849 T^{6} + 21646 p T^{7} - 5128 p^{2} T^{8} - 1007 p^{3} T^{9} - 5 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} ) \)
43 \( 1 - 23 T + 113 T^{2} + 896 T^{3} - 1332 T^{4} - 148755 T^{5} + 870961 T^{6} + 3952150 T^{7} - 29247526 T^{8} - 288761288 T^{9} + 65334263 p T^{10} + 289322302 T^{11} - 86189636971 T^{12} + 289322302 p T^{13} + 65334263 p^{3} T^{14} - 288761288 p^{3} T^{15} - 29247526 p^{4} T^{16} + 3952150 p^{5} T^{17} + 870961 p^{6} T^{18} - 148755 p^{7} T^{19} - 1332 p^{8} T^{20} + 896 p^{9} T^{21} + 113 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \)
47 \( ( 1 + T + 220 T^{2} + 76 T^{3} + 21464 T^{4} - 167 T^{5} + 1253383 T^{6} - 167 p T^{7} + 21464 p^{2} T^{8} + 76 p^{3} T^{9} + 220 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \)
53 \( 1 + 4 T - 69 T^{2} + 608 T^{3} + 3817 T^{4} - 53819 T^{5} + 247674 T^{6} + 1787013 T^{7} - 23754551 T^{8} + 62188347 T^{9} + 305246782 T^{10} - 5161623099 T^{11} + 3988104733 T^{12} - 5161623099 p T^{13} + 305246782 p^{2} T^{14} + 62188347 p^{3} T^{15} - 23754551 p^{4} T^{16} + 1787013 p^{5} T^{17} + 247674 p^{6} T^{18} - 53819 p^{7} T^{19} + 3817 p^{8} T^{20} + 608 p^{9} T^{21} - 69 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
59 \( ( 1 + 11 T + 151 T^{2} + 725 T^{3} + 10547 T^{4} + 66515 T^{5} + 880441 T^{6} + 66515 p T^{7} + 10547 p^{2} T^{8} + 725 p^{3} T^{9} + 151 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 25 T + 500 T^{2} - 6951 T^{3} + 1357 p T^{4} - 796808 T^{5} + 6827935 T^{6} - 796808 p T^{7} + 1357 p^{3} T^{8} - 6951 p^{3} T^{9} + 500 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 2 T + 177 T^{2} - 243 T^{3} + 18145 T^{4} - 40274 T^{5} + 1465189 T^{6} - 40274 p T^{7} + 18145 p^{2} T^{8} - 243 p^{3} T^{9} + 177 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 11 T + 244 T^{2} - 2045 T^{3} + 18530 T^{4} - 153299 T^{5} + 887101 T^{6} - 153299 p T^{7} + 18530 p^{2} T^{8} - 2045 p^{3} T^{9} + 244 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 - 24 T + 42 T^{2} + 3412 T^{3} - 16221 T^{4} - 351231 T^{5} + 2912248 T^{6} + 10402335 T^{7} - 128561004 T^{8} + 12356683 T^{9} - 873975840 T^{10} - 31448077707 T^{11} + 794156097013 T^{12} - 31448077707 p T^{13} - 873975840 p^{2} T^{14} + 12356683 p^{3} T^{15} - 128561004 p^{4} T^{16} + 10402335 p^{5} T^{17} + 2912248 p^{6} T^{18} - 351231 p^{7} T^{19} - 16221 p^{8} T^{20} + 3412 p^{9} T^{21} + 42 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
79 \( ( 1 - T + 179 T^{2} + 915 T^{3} + 21280 T^{4} + 88630 T^{5} + 2361307 T^{6} + 88630 p T^{7} + 21280 p^{2} T^{8} + 915 p^{3} T^{9} + 179 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 4 T - 294 T^{2} + 1948 T^{3} + 44977 T^{4} - 398395 T^{5} - 3801708 T^{6} + 52335429 T^{7} + 120574846 T^{8} - 4218193263 T^{9} + 14917345312 T^{10} + 151178074263 T^{11} - 2170254522125 T^{12} + 151178074263 p T^{13} + 14917345312 p^{2} T^{14} - 4218193263 p^{3} T^{15} + 120574846 p^{4} T^{16} + 52335429 p^{5} T^{17} - 3801708 p^{6} T^{18} - 398395 p^{7} T^{19} + 44977 p^{8} T^{20} + 1948 p^{9} T^{21} - 294 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 2 T - 255 T^{2} + 326 T^{3} + 33115 T^{4} + 1237 T^{5} - 2035002 T^{6} - 10969065 T^{7} + 3238993 T^{8} + 1541883639 T^{9} + 12567967198 T^{10} - 74858833077 T^{11} - 1448697683687 T^{12} - 74858833077 p T^{13} + 12567967198 p^{2} T^{14} + 1541883639 p^{3} T^{15} + 3238993 p^{4} T^{16} - 10969065 p^{5} T^{17} - 2035002 p^{6} T^{18} + 1237 p^{7} T^{19} + 33115 p^{8} T^{20} + 326 p^{9} T^{21} - 255 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
97 \( ( 1 + 18 T - 36 T^{2} - 490 T^{3} + 31950 T^{4} + 166842 T^{5} - 1649145 T^{6} + 166842 p T^{7} + 31950 p^{2} T^{8} - 490 p^{3} T^{9} - 36 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.30013868776587544154278398001, −3.10720554267598179695983736781, −2.96714645006397097404559514240, −2.80028010869062563103418064514, −2.79528406747245815343377405370, −2.65802543077249008264480254044, −2.59027166148629034227906074434, −2.49131979764861365838878050463, −2.47802009485665791030683572322, −2.41367702432752893146714146140, −2.40648433876455633147101083518, −2.28239644659945936398164086706, −2.25010280625667741002585228036, −1.75074932614824421364297080782, −1.63142191599702669803506430823, −1.62850325287363171828606148187, −1.55854703639707183198003228307, −1.43755622924931846795747316274, −1.10280953856654043222127451424, −1.02566870571423921519659894992, −0.886672884201509024030991763972, −0.61286076689001963247578829528, −0.56323877189305904094591506741, −0.33296095607326357615030538926, −0.32459048819994495453110198454, 0.32459048819994495453110198454, 0.33296095607326357615030538926, 0.56323877189305904094591506741, 0.61286076689001963247578829528, 0.886672884201509024030991763972, 1.02566870571423921519659894992, 1.10280953856654043222127451424, 1.43755622924931846795747316274, 1.55854703639707183198003228307, 1.62850325287363171828606148187, 1.63142191599702669803506430823, 1.75074932614824421364297080782, 2.25010280625667741002585228036, 2.28239644659945936398164086706, 2.40648433876455633147101083518, 2.41367702432752893146714146140, 2.47802009485665791030683572322, 2.49131979764861365838878050463, 2.59027166148629034227906074434, 2.65802543077249008264480254044, 2.79528406747245815343377405370, 2.80028010869062563103418064514, 2.96714645006397097404559514240, 3.10720554267598179695983736781, 3.30013868776587544154278398001

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

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