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  + 1.67·2-s − 3-s + 0.820·4-s + 3.59·5-s − 1.67·6-s + 1.09·7-s − 1.98·8-s + 9-s + 6.03·10-s − 1.52·11-s − 0.820·12-s − 0.0461·13-s + 1.83·14-s − 3.59·15-s − 4.96·16-s − 17-s + 1.67·18-s − 1.91·19-s + 2.94·20-s − 1.09·21-s − 2.56·22-s − 5.70·23-s + 1.98·24-s + 7.91·25-s − 0.0775·26-s − 27-s + 0.898·28-s + ⋯
L(s)  = 1  + 1.18·2-s − 0.577·3-s + 0.410·4-s + 1.60·5-s − 0.685·6-s + 0.413·7-s − 0.700·8-s + 0.333·9-s + 1.90·10-s − 0.461·11-s − 0.236·12-s − 0.0128·13-s + 0.491·14-s − 0.927·15-s − 1.24·16-s − 0.242·17-s + 0.395·18-s − 0.439·19-s + 0.659·20-s − 0.238·21-s − 0.547·22-s − 1.18·23-s + 0.404·24-s + 1.58·25-s − 0.0152·26-s − 0.192·27-s + 0.169·28-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 \( 1 - 1.67T + 2T^{2} \)
5 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 + 1.52T + 11T^{2} \)
13 \( 1 + 0.0461T + 13T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + 8.96T + 29T^{2} \)
31 \( 1 - 1.93T + 31T^{2} \)
37 \( 1 + 6.02T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 0.305T + 43T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 - 0.193T + 53T^{2} \)
59 \( 1 + 2.32T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 4.82T + 67T^{2} \)
71 \( 1 + 8.61T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 - 4.00T + 79T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 + 2.17T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−7.17043562972879735976570668890, −6.36148696647921922345137747609, −5.92780775925344641175688663556, −5.40951256226008972657126191445, −4.85448572959677447875596767483, −4.13527511451041056284037075649, −3.19286407035944654754549770306, −2.20717697481632822947199708072, −1.67122573749126125240337784963, 0, 1.67122573749126125240337784963, 2.20717697481632822947199708072, 3.19286407035944654754549770306, 4.13527511451041056284037075649, 4.85448572959677447875596767483, 5.40951256226008972657126191445, 5.92780775925344641175688663556, 6.36148696647921922345137747609, 7.17043562972879735976570668890

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