Properties

Label 4-1400e2-1.1-c3e2-0-0
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $6823.20$
Root an. cond. $9.08860$
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  + 2·3-s − 14·7-s + 6·9-s + 36·11-s − 42·13-s − 8·17-s − 118·19-s − 28·21-s + 104·23-s + 70·27-s − 56·29-s + 20·31-s + 72·33-s − 504·37-s − 84·39-s − 544·41-s − 412·43-s + 500·47-s + 147·49-s − 16·51-s − 268·53-s − 236·57-s − 198·59-s − 346·61-s − 84·63-s − 1.00e3·67-s + 208·69-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.755·7-s + 2/9·9-s + 0.986·11-s − 0.896·13-s − 0.114·17-s − 1.42·19-s − 0.290·21-s + 0.942·23-s + 0.498·27-s − 0.358·29-s + 0.115·31-s + 0.379·33-s − 2.23·37-s − 0.344·39-s − 2.07·41-s − 1.46·43-s + 1.55·47-s + 3/7·49-s − 0.0439·51-s − 0.694·53-s − 0.548·57-s − 0.436·59-s − 0.726·61-s − 0.167·63-s − 1.83·67-s + 0.362·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6823.20\)
Root analytic conductor: \(9.08860\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3060612080\)
\(L(\frac12)\) \(\approx\) \(0.3060612080\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 36 T + 934 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 42 T + 3410 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 7790 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 118 T + 7566 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 104 T + 26126 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 56 T + 21974 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 20 T + 48510 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 504 T + 159110 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 544 T + 193358 T^{2} + 544 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 412 T + 190278 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 500 T + 258974 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 268 T + 20222 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 198 T + 410926 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 346 T + 429114 T^{2} + 346 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1008 T + 591974 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1224 T + 980014 T^{2} + 1224 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 716 T + 832326 T^{2} + 716 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 584 T + 997470 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1230 T + 1395886 T^{2} - 1230 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 596 T + 39542 T^{2} - 596 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 856 T + 1729230 T^{2} - 856 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.247191865916562790386834133366, −8.897408909632159974529335123639, −8.751124551332377281709999950807, −8.432277342197930653850807450743, −7.63932789439115357425513726881, −7.31666557899714357951138525389, −6.95066348183951728273534919858, −6.66161130944813201288728020808, −6.10322859091422047069581511712, −5.89485236981368588727122681496, −5.01864735230323787152976423778, −4.72427654915745909764995099118, −4.43297409756764259179233976796, −3.58719627559626712766070286143, −3.35050298206753065558402183842, −2.98953597654980956862581602505, −2.02719198387629948427155101349, −1.89613501645320105158382419942, −1.09764694896281329126278469147, −0.12549630669852939249034445158, 0.12549630669852939249034445158, 1.09764694896281329126278469147, 1.89613501645320105158382419942, 2.02719198387629948427155101349, 2.98953597654980956862581602505, 3.35050298206753065558402183842, 3.58719627559626712766070286143, 4.43297409756764259179233976796, 4.72427654915745909764995099118, 5.01864735230323787152976423778, 5.89485236981368588727122681496, 6.10322859091422047069581511712, 6.66161130944813201288728020808, 6.95066348183951728273534919858, 7.31666557899714357951138525389, 7.63932789439115357425513726881, 8.432277342197930653850807450743, 8.751124551332377281709999950807, 8.897408909632159974529335123639, 9.247191865916562790386834133366

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