Properties

Label 4-1617e2-1.1-c3e2-0-0
Degree $4$
Conductor $2614689$
Sign $1$
Analytic cond. $9102.32$
Root an. cond. $9.76760$
Motivic weight $3$
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 + 9·4-s + 14·5-s + 6·6-s + 25·8-s + 27·9-s + 14·10-s − 22·11-s + 54·12-s − 30·13-s + 84·15-s + 41·16-s − 106·17-s + 27·18-s − 50·19-s + 126·20-s − 22·22-s + 134·23-s + 150·24-s − 6·25-s − 30·26-s + 108·27-s − 198·29-s + 84·30-s − 360·31-s + 249·32-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.15·3-s + 9/8·4-s + 1.25·5-s + 0.408·6-s + 1.10·8-s + 9-s + 0.442·10-s − 0.603·11-s + 1.29·12-s − 0.640·13-s + 1.44·15-s + 0.640·16-s − 1.51·17-s + 0.353·18-s − 0.603·19-s + 1.40·20-s − 0.213·22-s + 1.21·23-s + 1.27·24-s − 0.0479·25-s − 0.226·26-s + 0.769·27-s − 1.26·29-s + 0.511·30-s − 2.08·31-s + 1.37·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2614689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2614689\)    =    \(3^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(9102.32\)
Root analytic conductor: \(9.76760\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2614689,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.991177917\)
\(L(\frac12)\) \(\approx\) \(8.991177917\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 - T - p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 14 T + 202 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.247993038261810193896109912204, −8.947355632769483243840650400036, −8.527791580726799853797526746065, −7.889090627620246576375429829966, −7.48612524589023936467569421851, −7.18021327818016079422293239077, −7.12510630542045740798318106003, −6.28821728140332338664502453513, −6.14483887649982209579735570750, −5.60743907355253857025836877960, −4.99417356579414459675528302710, −4.80610676343150286493823471873, −4.01917259563620011544487403920, −3.82432704759850637941606073354, −3.00375801437884847587761137736, −2.50932926250790549815058638282, −2.29391629681456749485592140702, −1.81995318623054412485250398184, −1.56996547654290206972112084446, −0.47370988602242570374313802893, 0.47370988602242570374313802893, 1.56996547654290206972112084446, 1.81995318623054412485250398184, 2.29391629681456749485592140702, 2.50932926250790549815058638282, 3.00375801437884847587761137736, 3.82432704759850637941606073354, 4.01917259563620011544487403920, 4.80610676343150286493823471873, 4.99417356579414459675528302710, 5.60743907355253857025836877960, 6.14483887649982209579735570750, 6.28821728140332338664502453513, 7.12510630542045740798318106003, 7.18021327818016079422293239077, 7.48612524589023936467569421851, 7.889090627620246576375429829966, 8.527791580726799853797526746065, 8.947355632769483243840650400036, 9.247993038261810193896109912204

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