Properties

Degree 4
Conductor $ 17^{2} \cdot 43^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s − 4-s − 5-s − 2·6-s + 8·8-s − 9-s + 2·10-s − 4·11-s − 12-s − 5·13-s − 15-s − 7·16-s − 2·17-s + 2·18-s + 6·19-s + 20-s + 8·22-s + 4·23-s + 8·24-s − 5·25-s + 10·26-s − 2·29-s + 2·30-s − 12·31-s − 14·32-s − 4·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 2.82·8-s − 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.38·13-s − 0.258·15-s − 7/4·16-s − 0.485·17-s + 0.471·18-s + 1.37·19-s + 0.223·20-s + 1.70·22-s + 0.834·23-s + 1.63·24-s − 25-s + 1.96·26-s − 0.371·29-s + 0.365·30-s − 2.15·31-s − 2.47·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(534361\)    =    \(17^{2} \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{731} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 534361,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad17$C_1$ \( ( 1 + T )^{2} \)
43$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 7 T + 82 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_4$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_4$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 114 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 142 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 7 T + 150 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 3 T + 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 11 T + 90 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−9.888980249552727850051448226179, −9.601340392971392986442761877741, −9.308738138518385212077246178636, −9.073808570621472557868833840016, −8.325681979355988572320170938278, −8.189016891308197204101410067501, −7.72802013924908393346942829063, −7.45534630212084004733258656458, −7.13133553726181703840629357325, −6.37865224481598929073326882895, −5.31308685634011262394320915221, −5.13202739100140154387529971659, −4.91376999066276333896195008106, −4.18682650965189249457311233395, −3.30663455735130281510159129433, −3.23433596067551510250460117016, −2.05778752726431189114135828943, −1.54754060973099887411111284050, 0, 0, 1.54754060973099887411111284050, 2.05778752726431189114135828943, 3.23433596067551510250460117016, 3.30663455735130281510159129433, 4.18682650965189249457311233395, 4.91376999066276333896195008106, 5.13202739100140154387529971659, 5.31308685634011262394320915221, 6.37865224481598929073326882895, 7.13133553726181703840629357325, 7.45534630212084004733258656458, 7.72802013924908393346942829063, 8.189016891308197204101410067501, 8.325681979355988572320170938278, 9.073808570621472557868833840016, 9.308738138518385212077246178636, 9.601340392971392986442761877741, 9.888980249552727850051448226179

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