Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.56·3-s − 4-s − 2.56·5-s − 2.56·6-s + 3·8-s + 3.56·9-s + 2.56·10-s − 2·11-s − 2.56·12-s − 4.56·13-s − 6.56·15-s − 16-s − 17-s − 3.56·18-s − 1.12·19-s + 2.56·20-s + 2·22-s + 2·23-s + 7.68·24-s + 1.56·25-s + 4.56·26-s + 1.43·27-s − 5.12·29-s + 6.56·30-s − 6·31-s − 5·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.47·3-s − 0.5·4-s − 1.14·5-s − 1.04·6-s + 1.06·8-s + 1.18·9-s + 0.810·10-s − 0.603·11-s − 0.739·12-s − 1.26·13-s − 1.69·15-s − 0.250·16-s − 0.242·17-s − 0.839·18-s − 0.257·19-s + 0.572·20-s + 0.426·22-s + 0.417·23-s + 1.56·24-s + 0.312·25-s + 0.894·26-s + 0.276·27-s − 0.951·29-s + 1.19·30-s − 1.07·31-s − 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{731} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4.56T + 13T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 1.43T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 8.80T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 1.43T + 67T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 - 8.80T + 73T^{2} \)
79 \( 1 + 0.876T + 79T^{2} \)
83 \( 1 + 4.80T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 0.246T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.601340392971392986442761877741, −9.073808570621472557868833840016, −8.189016891308197204101410067501, −7.72802013924908393346942829063, −7.13133553726181703840629357325, −5.13202739100140154387529971659, −4.18682650965189249457311233395, −3.30663455735130281510159129433, −2.05778752726431189114135828943, 0, 2.05778752726431189114135828943, 3.30663455735130281510159129433, 4.18682650965189249457311233395, 5.13202739100140154387529971659, 7.13133553726181703840629357325, 7.72802013924908393346942829063, 8.189016891308197204101410067501, 9.073808570621472557868833840016, 9.601340392971392986442761877741

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