Properties

Degree 4
Conductor $ 2^{2} \cdot 3019^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 4

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 5·5-s − 6·6-s − 6·7-s + 4·8-s + 2·9-s − 10·10-s − 7·11-s − 9·12-s − 6·13-s − 12·14-s + 15·15-s + 5·16-s − 6·17-s + 4·18-s − 6·19-s − 15·20-s + 18·21-s − 14·22-s − 6·23-s − 12·24-s + 10·25-s − 12·26-s + 6·27-s − 18·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.23·5-s − 2.44·6-s − 2.26·7-s + 1.41·8-s + 2/3·9-s − 3.16·10-s − 2.11·11-s − 2.59·12-s − 1.66·13-s − 3.20·14-s + 3.87·15-s + 5/4·16-s − 1.45·17-s + 0.942·18-s − 1.37·19-s − 3.35·20-s + 3.92·21-s − 2.98·22-s − 1.25·23-s − 2.44·24-s + 2·25-s − 2.35·26-s + 1.15·27-s − 3.40·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36457444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36457444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36457444\)    =    \(2^{2} \cdot 3019^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6038} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(4,\ 36457444,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3019\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3019\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
3019 \( 1+O(T) \)
good3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T - 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 24 T + 257 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 133 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 149 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 32 T + 445 T^{2} - 32 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.49637706986989643634034703940, −7.20193873697552341772487463710, −6.46178064959430146484122118225, −6.45082350381719454975173900019, −6.04002451607903648633524029033, −5.93825411806962806963479187343, −5.25737649908674734961429262711, −4.99050022039682986454604642705, −4.56388580421297763359868069522, −4.49115166715489506444535199749, −3.94922063529863278727306577464, −3.37692936114172083921266340471, −3.22422217533988060994973477349, −2.89493040810100742624055535855, −2.20540295289471703522520176571, −1.94434150521789333176918400896, 0, 0, 0, 0, 1.94434150521789333176918400896, 2.20540295289471703522520176571, 2.89493040810100742624055535855, 3.22422217533988060994973477349, 3.37692936114172083921266340471, 3.94922063529863278727306577464, 4.49115166715489506444535199749, 4.56388580421297763359868069522, 4.99050022039682986454604642705, 5.25737649908674734961429262711, 5.93825411806962806963479187343, 6.04002451607903648633524029033, 6.45082350381719454975173900019, 6.46178064959430146484122118225, 7.20193873697552341772487463710, 7.49637706986989643634034703940

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