Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70·2-s − 3-s + 0.912·4-s + 1.58·5-s + 1.70·6-s − 2.98·7-s + 1.85·8-s + 9-s − 2.70·10-s − 5.30·11-s − 0.912·12-s − 4.73·13-s + 5.10·14-s − 1.58·15-s − 4.99·16-s − 17-s − 1.70·18-s + 3.98·19-s + 1.44·20-s + 2.98·21-s + 9.05·22-s + 2.22·23-s − 1.85·24-s − 2.49·25-s + 8.08·26-s − 27-s − 2.72·28-s + ⋯
L(s)  = 1  − 1.20·2-s − 0.577·3-s + 0.456·4-s + 0.708·5-s + 0.696·6-s − 1.12·7-s + 0.656·8-s + 0.333·9-s − 0.854·10-s − 1.60·11-s − 0.263·12-s − 1.31·13-s + 1.36·14-s − 0.408·15-s − 1.24·16-s − 0.242·17-s − 0.402·18-s + 0.913·19-s + 0.323·20-s + 0.652·21-s + 1.93·22-s + 0.463·23-s − 0.378·24-s − 0.498·25-s + 1.58·26-s − 0.192·27-s − 0.515·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 1.70T + 2T^{2} \)
5 \( 1 - 1.58T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 + 5.30T + 11T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
19 \( 1 - 3.98T + 19T^{2} \)
23 \( 1 - 2.22T + 23T^{2} \)
29 \( 1 + 4.49T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 - 5.16T + 37T^{2} \)
41 \( 1 - 5.33T + 41T^{2} \)
43 \( 1 - 8.14T + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 - 6.35T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 8.69T + 61T^{2} \)
67 \( 1 - 5.45T + 67T^{2} \)
71 \( 1 + 5.81T + 71T^{2} \)
73 \( 1 - 5.83T + 73T^{2} \)
79 \( 1 - 6.64T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + 9.51T + 97T^{2} \)
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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

−7.60782876574603554204908997924, −6.97990958304663323143963860407, −6.23648623097462018891968232712, −5.34233489692069236173664789328, −5.02850006855162425758797690873, −3.89090577100162354635392108207, −2.66132879127184249019338877688, −2.20759778844057303075597984061, −0.826474535958942925243616333311, 0, 0.826474535958942925243616333311, 2.20759778844057303075597984061, 2.66132879127184249019338877688, 3.89090577100162354635392108207, 5.02850006855162425758797690873, 5.34233489692069236173664789328, 6.23648623097462018891968232712, 6.97990958304663323143963860407, 7.60782876574603554204908997924

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