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  − 2.76·2-s + 3-s + 5.62·4-s − 0.677·5-s − 2.76·6-s + 3.14·7-s − 9.99·8-s + 9-s + 1.86·10-s − 2.07·11-s + 5.62·12-s + 5.11·13-s − 8.67·14-s − 0.677·15-s + 16.3·16-s − 17-s − 2.76·18-s + 1.32·19-s − 3.80·20-s + 3.14·21-s + 5.72·22-s + 1.26·23-s − 9.99·24-s − 4.54·25-s − 14.1·26-s + 27-s + 17.6·28-s + ⋯
L(s)  = 1  − 1.95·2-s + 0.577·3-s + 2.81·4-s − 0.302·5-s − 1.12·6-s + 1.18·7-s − 3.53·8-s + 0.333·9-s + 0.591·10-s − 0.625·11-s + 1.62·12-s + 1.41·13-s − 2.31·14-s − 0.174·15-s + 4.08·16-s − 0.242·17-s − 0.650·18-s + 0.302·19-s − 0.851·20-s + 0.686·21-s + 1.22·22-s + 0.263·23-s − 2.04·24-s − 0.908·25-s − 2.77·26-s + 0.192·27-s + 3.33·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 2.76T + 2T^{2} \)
5 \( 1 + 0.677T + 5T^{2} \)
7 \( 1 - 3.14T + 7T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 - 5.11T + 13T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 - 1.26T + 23T^{2} \)
29 \( 1 - 2.79T + 29T^{2} \)
31 \( 1 + 1.04T + 31T^{2} \)
37 \( 1 - 1.14T + 37T^{2} \)
41 \( 1 + 8.84T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 + 6.36T + 53T^{2} \)
59 \( 1 + 4.43T + 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 6.45T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 0.815T + 73T^{2} \)
79 \( 1 + 5.63T + 79T^{2} \)
83 \( 1 - 3.91T + 83T^{2} \)
89 \( 1 - 6.79T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.88578562528193018332625221567, −7.20292499739211738812536628208, −6.44735398679393799179168029636, −5.69569611272620702619351148638, −4.66733793595197221640642088172, −3.49372148376215033351693216852, −2.82366069794294298632799116102, −1.70069029315566022738981080928, −1.40658309437981031733246350617, 0, 1.40658309437981031733246350617, 1.70069029315566022738981080928, 2.82366069794294298632799116102, 3.49372148376215033351693216852, 4.66733793595197221640642088172, 5.69569611272620702619351148638, 6.44735398679393799179168029636, 7.20292499739211738812536628208, 7.88578562528193018332625221567

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