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  − 0.212·2-s − 2.83·3-s − 1.95·4-s − 1.65·5-s + 0.602·6-s + 2.53·7-s + 0.839·8-s + 5.06·9-s + 0.352·10-s + 1.11·11-s + 5.55·12-s + 2.39·13-s − 0.539·14-s + 4.71·15-s + 3.73·16-s − 17-s − 1.07·18-s − 5.01·19-s + 3.24·20-s − 7.21·21-s − 0.236·22-s + 3.24·23-s − 2.38·24-s − 2.24·25-s − 0.508·26-s − 5.85·27-s − 4.96·28-s + ⋯
L(s)  = 1  − 0.150·2-s − 1.63·3-s − 0.977·4-s − 0.741·5-s + 0.246·6-s + 0.959·7-s + 0.296·8-s + 1.68·9-s + 0.111·10-s + 0.336·11-s + 1.60·12-s + 0.664·13-s − 0.144·14-s + 1.21·15-s + 0.932·16-s − 0.242·17-s − 0.253·18-s − 1.15·19-s + 0.725·20-s − 1.57·21-s − 0.0504·22-s + 0.677·23-s − 0.486·24-s − 0.449·25-s − 0.0997·26-s − 1.12·27-s − 0.938·28-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 \( 1 + 0.212T + 2T^{2} \)
3 \( 1 + 2.83T + 3T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 - 2.53T + 7T^{2} \)
11 \( 1 - 1.11T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 - 0.498T + 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
47 \( 1 - 0.400T + 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 - 0.0130T + 61T^{2} \)
67 \( 1 + 8.24T + 67T^{2} \)
71 \( 1 + 8.15T + 71T^{2} \)
73 \( 1 - 5.99T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 + 1.09T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 6.18T + 97T^{2} \)
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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

−10.25750350205496635939165222134, −9.026830485771565425339356669431, −8.282061455517674885545496171180, −7.32746757927295565065801806840, −6.26418066279004017806590276145, −5.33597653404152324270618841383, −4.55853515132601337156290026062, −3.88745321227365620147515252028, −1.36727566051240818908008943907, 0, 1.36727566051240818908008943907, 3.88745321227365620147515252028, 4.55853515132601337156290026062, 5.33597653404152324270618841383, 6.26418066279004017806590276145, 7.32746757927295565065801806840, 8.282061455517674885545496171180, 9.026830485771565425339356669431, 10.25750350205496635939165222134

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