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  + 2.31·2-s + 3-s + 3.37·4-s + 0.731·5-s + 2.31·6-s − 2.06·7-s + 3.18·8-s + 9-s + 1.69·10-s − 4.53·11-s + 3.37·12-s − 5.27·13-s − 4.79·14-s + 0.731·15-s + 0.639·16-s − 17-s + 2.31·18-s + 4.22·19-s + 2.46·20-s − 2.06·21-s − 10.5·22-s + 1.41·23-s + 3.18·24-s − 4.46·25-s − 12.2·26-s + 27-s − 6.97·28-s + ⋯
L(s)  = 1  + 1.63·2-s + 0.577·3-s + 1.68·4-s + 0.327·5-s + 0.946·6-s − 0.781·7-s + 1.12·8-s + 0.333·9-s + 0.536·10-s − 1.36·11-s + 0.974·12-s − 1.46·13-s − 1.28·14-s + 0.188·15-s + 0.159·16-s − 0.242·17-s + 0.546·18-s + 0.970·19-s + 0.552·20-s − 0.451·21-s − 2.24·22-s + 0.295·23-s + 0.650·24-s − 0.892·25-s − 2.39·26-s + 0.192·27-s − 1.31·28-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 \( 1 - 2.31T + 2T^{2} \)
5 \( 1 - 0.731T + 5T^{2} \)
7 \( 1 + 2.06T + 7T^{2} \)
11 \( 1 + 4.53T + 11T^{2} \)
13 \( 1 + 5.27T + 13T^{2} \)
19 \( 1 - 4.22T + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 - 3.67T + 29T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 + 6.37T + 37T^{2} \)
41 \( 1 - 5.71T + 41T^{2} \)
43 \( 1 + 7.14T + 43T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 - 9.76T + 53T^{2} \)
59 \( 1 + 4.22T + 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 + 0.501T + 67T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 8.98T + 83T^{2} \)
89 \( 1 - 5.58T + 89T^{2} \)
97 \( 1 + 7.11T + 97T^{2} \)
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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

−7.27004933135900648550176571133, −6.78250311462090506666257270062, −5.76553793813886815995393656766, −5.36034650223284307130036886523, −4.70953858873447740972429539623, −3.90382108710026632951664400122, −3.00261831726748386217574622232, −2.71192355233127980510969125599, −1.88828911613691814616385902589, 0, 1.88828911613691814616385902589, 2.71192355233127980510969125599, 3.00261831726748386217574622232, 3.90382108710026632951664400122, 4.70953858873447740972429539623, 5.36034650223284307130036886523, 5.76553793813886815995393656766, 6.78250311462090506666257270062, 7.27004933135900648550176571133

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