Properties

Label 4-1840e2-1.1-c1e2-0-0
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $215.868$
Root an. cond. $3.83307$
Motivic weight $1$
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  + 3-s − 2·5-s − 7-s − 3·11-s + 7·13-s − 2·15-s − 3·17-s − 7·19-s − 21-s − 2·23-s + 3·25-s + 2·27-s + 6·29-s − 7·31-s − 3·33-s + 2·35-s − 8·37-s + 7·39-s − 9·41-s − 4·43-s + 18·47-s − 8·49-s − 3·51-s + 12·53-s + 6·55-s − 7·57-s + 18·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s − 0.904·11-s + 1.94·13-s − 0.516·15-s − 0.727·17-s − 1.60·19-s − 0.218·21-s − 0.417·23-s + 3/5·25-s + 0.384·27-s + 1.11·29-s − 1.25·31-s − 0.522·33-s + 0.338·35-s − 1.31·37-s + 1.12·39-s − 1.40·41-s − 0.609·43-s + 2.62·47-s − 8/7·49-s − 0.420·51-s + 1.64·53-s + 0.809·55-s − 0.927·57-s + 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(215.868\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3385600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.290483981\)
\(L(\frac12)\) \(\approx\) \(1.290483981\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 18 T + 154 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 18 T + 178 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
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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

−9.192558564449213104883704394554, −8.744147764448629160552381936385, −8.572613416526234713200387429236, −8.515253642608303670110376157663, −8.105654653243987316633960702933, −7.45851576494995521440588353022, −7.02393613300488824488128644459, −6.84408143774689620859325903109, −6.28370926587918452379634881985, −5.91050763643489754797739481778, −5.43262524105306635957983829240, −4.90763234032842049286194293711, −4.35724510353962739433974942985, −3.97925541850641924941011032608, −3.43266895891183257743930666625, −3.40263296772532377232184229017, −2.31734844278758075461923703741, −2.31289659133432238693076633442, −1.32419911688698329532723629697, −0.41599783352405562787186114485, 0.41599783352405562787186114485, 1.32419911688698329532723629697, 2.31289659133432238693076633442, 2.31734844278758075461923703741, 3.40263296772532377232184229017, 3.43266895891183257743930666625, 3.97925541850641924941011032608, 4.35724510353962739433974942985, 4.90763234032842049286194293711, 5.43262524105306635957983829240, 5.91050763643489754797739481778, 6.28370926587918452379634881985, 6.84408143774689620859325903109, 7.02393613300488824488128644459, 7.45851576494995521440588353022, 8.105654653243987316633960702933, 8.515253642608303670110376157663, 8.572613416526234713200387429236, 8.744147764448629160552381936385, 9.192558564449213104883704394554

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