Properties

Label 4-2475e2-1.1-c3e2-0-4
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $21324.6$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 9·4-s − 24·7-s + 25·8-s + 22·11-s − 30·13-s − 24·14-s + 41·16-s + 106·17-s + 50·19-s + 22·22-s + 134·23-s − 30·26-s − 216·28-s + 198·29-s + 360·31-s + 249·32-s + 106·34-s + 328·37-s + 50·38-s + 782·41-s − 386·43-s + 198·44-s + 134·46-s + 266·47-s + 134·49-s − 270·52-s + ⋯
L(s)  = 1  + 0.353·2-s + 9/8·4-s − 1.29·7-s + 1.10·8-s + 0.603·11-s − 0.640·13-s − 0.458·14-s + 0.640·16-s + 1.51·17-s + 0.603·19-s + 0.213·22-s + 1.21·23-s − 0.226·26-s − 1.45·28-s + 1.26·29-s + 2.08·31-s + 1.37·32-s + 0.534·34-s + 1.45·37-s + 0.213·38-s + 2.97·41-s − 1.36·43-s + 0.678·44-s + 0.429·46-s + 0.825·47-s + 0.390·49-s − 0.720·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(21324.6\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.111274150\)
\(L(\frac12)\) \(\approx\) \(8.111274150\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 - T - p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 30 T + 4522 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 106 T + 7882 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 50 T + 14246 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 134 T + 26398 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 328 T + 62630 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 782 T + 285970 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 386 T + 179870 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 266 T + 92542 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 522 T + 295162 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 776 T + 528582 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1296 T + 1178926 T^{2} + 1296 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 652 T + 589506 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 324 T + 579670 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 452 T + 982470 T^{2} - 452 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.795759133585842790037926363210, −8.261579963786393068971411877193, −7.80968275803579131701313037068, −7.51235855780690021742990014837, −7.31943777769190295922530629347, −6.73778581700204188735748099078, −6.45017574975322343354199701980, −6.11976253167072302098022074699, −5.90904704424198216711065960583, −5.24107771260028231703855754604, −4.62111246498442470999945091099, −4.59061766979703980579529316556, −3.95696909420329897397791955117, −3.27799926686015587703525492087, −2.93943365083729515510651542215, −2.77023486848679734207366342686, −2.25404598355844444123514292016, −1.27496928258631777666068177414, −1.11018526135846204203044094696, −0.57126054679561116444234669376, 0.57126054679561116444234669376, 1.11018526135846204203044094696, 1.27496928258631777666068177414, 2.25404598355844444123514292016, 2.77023486848679734207366342686, 2.93943365083729515510651542215, 3.27799926686015587703525492087, 3.95696909420329897397791955117, 4.59061766979703980579529316556, 4.62111246498442470999945091099, 5.24107771260028231703855754604, 5.90904704424198216711065960583, 6.11976253167072302098022074699, 6.45017574975322343354199701980, 6.73778581700204188735748099078, 7.31943777769190295922530629347, 7.51235855780690021742990014837, 7.80968275803579131701313037068, 8.261579963786393068971411877193, 8.795759133585842790037926363210

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