Dirichlet series
| L(s) = 1 | + 12·11-s + 16·13-s + 4·17-s − 12·23-s − 4·29-s − 8·31-s − 20·37-s − 8·41-s + 8·47-s − 8·61-s + 32·67-s − 28·71-s − 24·79-s + 56·89-s − 44·97-s + 8·103-s + 44·107-s − 8·109-s + 32·113-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 192·143-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 4.43·13-s + 0.970·17-s − 2.50·23-s − 0.742·29-s − 1.43·31-s − 3.28·37-s − 1.24·41-s + 1.16·47-s − 1.02·61-s + 3.90·67-s − 3.32·71-s − 2.70·79-s + 5.93·89-s − 4.46·97-s + 0.788·103-s + 4.25·107-s − 0.766·109-s + 3.01·113-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 16.0·143-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.52879\times 10^{16}\) |
| Root analytic conductor: | \(4.94309\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.495637323\) |
| \(L(\frac12)\) | \(\approx\) | \(2.495637323\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 + T^{4} )^{3} \) | |
| 17 | \( 1 - 4 T - 30 T^{2} + 212 T^{3} - 57 T^{4} - 2384 T^{5} + 9980 T^{6} - 2384 p T^{7} - 57 p^{2} T^{8} + 212 p^{3} T^{9} - 30 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 - 44 T^{3} - 46 T^{4} - 36 T^{5} + 968 T^{6} + 1784 T^{7} + 843 T^{8} - 9088 T^{9} - 680 p^{2} T^{10} + 3904 p T^{11} + 61384 T^{12} + 3904 p^{2} T^{13} - 680 p^{4} T^{14} - 9088 p^{3} T^{15} + 843 p^{4} T^{16} + 1784 p^{5} T^{17} + 968 p^{6} T^{18} - 36 p^{7} T^{19} - 46 p^{8} T^{20} - 44 p^{9} T^{21} + p^{12} T^{24} \) |
| 11 | \( 1 - 12 T + 72 T^{2} - 32 p T^{3} + 134 p T^{4} - 4624 T^{5} + 11312 T^{6} - 19084 T^{7} - 34289 T^{8} + 466440 T^{9} - 2394104 T^{10} + 10254472 T^{11} - 37936256 T^{12} + 10254472 p T^{13} - 2394104 p^{2} T^{14} + 466440 p^{3} T^{15} - 34289 p^{4} T^{16} - 19084 p^{5} T^{17} + 11312 p^{6} T^{18} - 4624 p^{7} T^{19} + 134 p^{9} T^{20} - 32 p^{10} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 - 8 T + 76 T^{2} - 428 T^{3} + 2303 T^{4} - 9964 T^{5} + 38652 T^{6} - 9964 p T^{7} + 2303 p^{2} T^{8} - 428 p^{3} T^{9} + 76 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 124 T^{2} + 7506 T^{4} - 292204 T^{6} + 8306911 T^{8} - 189766264 T^{10} + 3786768444 T^{12} - 189766264 p^{2} T^{14} + 8306911 p^{4} T^{16} - 292204 p^{6} T^{18} + 7506 p^{8} T^{20} - 124 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 72 T^{2} + 80 T^{3} - 1946 T^{4} - 14656 T^{5} - 32560 T^{6} + 259892 T^{7} + 2779027 T^{8} + 12564288 T^{9} + 29320 p^{2} T^{10} - 8188120 p T^{11} - 1457658608 T^{12} - 8188120 p^{2} T^{13} + 29320 p^{4} T^{14} + 12564288 p^{3} T^{15} + 2779027 p^{4} T^{16} + 259892 p^{5} T^{17} - 32560 p^{6} T^{18} - 14656 p^{7} T^{19} - 1946 p^{8} T^{20} + 80 p^{9} T^{21} + 72 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 4 T + 8 T^{2} - 180 T^{3} + 506 T^{4} + 2316 T^{5} + 21416 T^{6} - 345628 T^{7} - 407169 T^{8} + 645416 T^{9} + 59897488 T^{10} - 128550792 T^{11} - 313824212 T^{12} - 128550792 p T^{13} + 59897488 p^{2} T^{14} + 645416 p^{3} T^{15} - 407169 p^{4} T^{16} - 345628 p^{5} T^{17} + 21416 p^{6} T^{18} + 2316 p^{7} T^{19} + 506 p^{8} T^{20} - 180 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 8 T + 32 T^{2} + 60 T^{3} - 2282 T^{4} - 13036 T^{5} - 29464 T^{6} + 142824 T^{7} + 4261575 T^{8} + 15592024 T^{9} + 14667032 T^{10} - 345273440 T^{11} - 4827009432 T^{12} - 345273440 p T^{13} + 14667032 p^{2} T^{14} + 15592024 p^{3} T^{15} + 4261575 p^{4} T^{16} + 142824 p^{5} T^{17} - 29464 p^{6} T^{18} - 13036 p^{7} T^{19} - 2282 p^{8} T^{20} + 60 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 20 T + 200 T^{2} + 1580 T^{3} + 194 p T^{4} - 13716 T^{5} - 461720 T^{6} - 4837388 T^{7} - 31278241 T^{8} - 84444504 T^{9} + 331975568 T^{10} + 6987157144 T^{11} + 58947110540 T^{12} + 6987157144 p T^{13} + 331975568 p^{2} T^{14} - 84444504 p^{3} T^{15} - 31278241 p^{4} T^{16} - 4837388 p^{5} T^{17} - 461720 p^{6} T^{18} - 13716 p^{7} T^{19} + 194 p^{9} T^{20} + 1580 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 8 T + 32 T^{2} + 360 T^{3} + 1030 T^{4} - 13880 T^{5} - 79200 T^{6} - 639128 T^{7} - 5607825 T^{8} - 6342448 T^{9} + 76527936 T^{10} + 887658896 T^{11} + 7855557012 T^{12} + 887658896 p T^{13} + 76527936 p^{2} T^{14} - 6342448 p^{3} T^{15} - 5607825 p^{4} T^{16} - 639128 p^{5} T^{17} - 79200 p^{6} T^{18} - 13880 p^{7} T^{19} + 1030 p^{8} T^{20} + 360 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 400 T^{2} + 76022 T^{4} - 9140920 T^{6} + 778282367 T^{8} - 49563290520 T^{10} + 2423044894788 T^{12} - 49563290520 p^{2} T^{14} + 778282367 p^{4} T^{16} - 9140920 p^{6} T^{18} + 76022 p^{8} T^{20} - 400 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 4 T + 4 p T^{2} - 712 T^{3} + 16447 T^{4} - 55252 T^{5} + 923372 T^{6} - 55252 p T^{7} + 16447 p^{2} T^{8} - 712 p^{3} T^{9} + 4 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 492 T^{2} + 114370 T^{4} - 16721148 T^{6} + 1726321023 T^{8} - 133687034040 T^{10} + 8013449956956 T^{12} - 133687034040 p^{2} T^{14} + 1726321023 p^{4} T^{16} - 16721148 p^{6} T^{18} + 114370 p^{8} T^{20} - 492 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 372 T^{2} + 72914 T^{4} - 9789956 T^{6} + 994030591 T^{8} - 80033516392 T^{10} + 5225462954684 T^{12} - 80033516392 p^{2} T^{14} + 994030591 p^{4} T^{16} - 9789956 p^{6} T^{18} + 72914 p^{8} T^{20} - 372 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 8 T + 32 T^{2} + 280 T^{3} + 7894 T^{4} + 52952 T^{5} + 210208 T^{6} + 1056392 T^{7} + 1537727 T^{8} + 42857232 T^{9} + 332015552 T^{10} - 2783679952 T^{11} - 109065974348 T^{12} - 2783679952 p T^{13} + 332015552 p^{2} T^{14} + 42857232 p^{3} T^{15} + 1537727 p^{4} T^{16} + 1056392 p^{5} T^{17} + 210208 p^{6} T^{18} + 52952 p^{7} T^{19} + 7894 p^{8} T^{20} + 280 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 16 T + 216 T^{2} - 1388 T^{3} + 16855 T^{4} - 157164 T^{5} + 1849812 T^{6} - 157164 p T^{7} + 16855 p^{2} T^{8} - 1388 p^{3} T^{9} + 216 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 28 T + 392 T^{2} + 3688 T^{3} + 13002 T^{4} - 153936 T^{5} - 2606320 T^{6} - 19094300 T^{7} - 12325409 T^{8} + 1060166688 T^{9} + 9831967880 T^{10} + 43542130544 T^{11} + 144697202768 T^{12} + 43542130544 p T^{13} + 9831967880 p^{2} T^{14} + 1060166688 p^{3} T^{15} - 12325409 p^{4} T^{16} - 19094300 p^{5} T^{17} - 2606320 p^{6} T^{18} - 153936 p^{7} T^{19} + 13002 p^{8} T^{20} + 3688 p^{9} T^{21} + 392 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 48 T^{3} - 3098 T^{4} - 14608 T^{5} + 1152 T^{6} - 2105856 T^{7} - 7751953 T^{8} + 168952448 T^{9} + 9184640 T^{10} + 5464652000 T^{11} + 122038602196 T^{12} + 5464652000 p T^{13} + 9184640 p^{2} T^{14} + 168952448 p^{3} T^{15} - 7751953 p^{4} T^{16} - 2105856 p^{5} T^{17} + 1152 p^{6} T^{18} - 14608 p^{7} T^{19} - 3098 p^{8} T^{20} + 48 p^{9} T^{21} + p^{12} T^{24} \) | |
| 79 | \( 1 + 24 T + 288 T^{2} + 1084 T^{3} - 5090 T^{4} + 118556 T^{5} + 4898792 T^{6} + 61864992 T^{7} + 297960775 T^{8} - 461305152 T^{9} + 2140238328 T^{10} + 361429508888 T^{11} + 5117697171064 T^{12} + 361429508888 p T^{13} + 2140238328 p^{2} T^{14} - 461305152 p^{3} T^{15} + 297960775 p^{4} T^{16} + 61864992 p^{5} T^{17} + 4898792 p^{6} T^{18} + 118556 p^{7} T^{19} - 5090 p^{8} T^{20} + 1084 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 640 T^{2} + 201470 T^{4} - 41544920 T^{6} + 6274166799 T^{8} - 732777674536 T^{10} + 67970550909332 T^{12} - 732777674536 p^{2} T^{14} + 6274166799 p^{4} T^{16} - 41544920 p^{6} T^{18} + 201470 p^{8} T^{20} - 640 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 28 T + 698 T^{2} - 11656 T^{3} + 172135 T^{4} - 2008188 T^{5} + 21025248 T^{6} - 2008188 p T^{7} + 172135 p^{2} T^{8} - 11656 p^{3} T^{9} + 698 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 44 T + 968 T^{2} + 16100 T^{3} + 253018 T^{4} + 3812372 T^{5} + 52427944 T^{6} + 672573820 T^{7} + 8224656847 T^{8} + 94725051416 T^{9} + 1036749534608 T^{10} + 11021110637128 T^{11} + 112097074521004 T^{12} + 11021110637128 p T^{13} + 1036749534608 p^{2} T^{14} + 94725051416 p^{3} T^{15} + 8224656847 p^{4} T^{16} + 672573820 p^{5} T^{17} + 52427944 p^{6} T^{18} + 3812372 p^{7} T^{19} + 253018 p^{8} T^{20} + 16100 p^{9} T^{21} + 968 p^{10} T^{22} + 44 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.66639056755275040340278410695, −2.44083621982238050273810174147, −2.41453786925053580303514029903, −2.35860563297396284232236241812, −2.22220006459809962791705350480, −2.21304587813733381947726457940, −2.01111677073716327656509848442, −1.83607373718543566761341483585, −1.80736589897550735206573181551, −1.72779728580975695835192002562, −1.72657021142163548428283130174, −1.72592335443193926025304215318, −1.63491685655964546858486939414, −1.55235291995348293206095601997, −1.38726197752063675866255233400, −1.29494096690614084314828956965, −1.22418313074834352638141960245, −0.992853028071862444546287556336, −0.960308336132632009480021196792, −0.70696293646702856414506075344, −0.69860963356326592234275151345, −0.69049787730078651867959080927, −0.65990767295626113189337212867, −0.14382885964471654176843211502, −0.081101991876935318651507061874, 0.081101991876935318651507061874, 0.14382885964471654176843211502, 0.65990767295626113189337212867, 0.69049787730078651867959080927, 0.69860963356326592234275151345, 0.70696293646702856414506075344, 0.960308336132632009480021196792, 0.992853028071862444546287556336, 1.22418313074834352638141960245, 1.29494096690614084314828956965, 1.38726197752063675866255233400, 1.55235291995348293206095601997, 1.63491685655964546858486939414, 1.72592335443193926025304215318, 1.72657021142163548428283130174, 1.72779728580975695835192002562, 1.80736589897550735206573181551, 1.83607373718543566761341483585, 2.01111677073716327656509848442, 2.21304587813733381947726457940, 2.22220006459809962791705350480, 2.35860563297396284232236241812, 2.41453786925053580303514029903, 2.44083621982238050273810174147, 2.66639056755275040340278410695
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.