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  + 2.71·2-s + 3-s + 5.38·4-s − 1.62·5-s + 2.71·6-s − 4.45·7-s + 9.18·8-s + 9-s − 4.42·10-s − 1.17·11-s + 5.38·12-s − 2.32·13-s − 12.1·14-s − 1.62·15-s + 14.2·16-s − 17-s + 2.71·18-s − 3.46·19-s − 8.76·20-s − 4.45·21-s − 3.19·22-s − 2.41·23-s + 9.18·24-s − 2.34·25-s − 6.31·26-s + 27-s − 23.9·28-s + ⋯
L(s)  = 1  + 1.92·2-s + 0.577·3-s + 2.69·4-s − 0.728·5-s + 1.10·6-s − 1.68·7-s + 3.24·8-s + 0.333·9-s − 1.39·10-s − 0.355·11-s + 1.55·12-s − 0.644·13-s − 3.23·14-s − 0.420·15-s + 3.55·16-s − 0.242·17-s + 0.640·18-s − 0.795·19-s − 1.96·20-s − 0.972·21-s − 0.682·22-s − 0.504·23-s + 1.87·24-s − 0.469·25-s − 1.23·26-s + 0.192·27-s − 4.53·28-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 \( 1 - 2.71T + 2T^{2} \)
5 \( 1 + 1.62T + 5T^{2} \)
7 \( 1 + 4.45T + 7T^{2} \)
11 \( 1 + 1.17T + 11T^{2} \)
13 \( 1 + 2.32T + 13T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 + 4.71T + 29T^{2} \)
31 \( 1 + 2.77T + 31T^{2} \)
37 \( 1 - 6.42T + 37T^{2} \)
41 \( 1 + 0.456T + 41T^{2} \)
43 \( 1 + 5.40T + 43T^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 - 3.91T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 9.67T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 + 4.21T + 79T^{2} \)
83 \( 1 - 7.58T + 83T^{2} \)
89 \( 1 - 2.41T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.24249633231614401445733581277, −6.67003473767429922270918013273, −6.02549729779034229189676976711, −5.36508946029324143081225691311, −4.33299014941150381173882608243, −3.96635753409583245577902504697, −3.25314866125894232870360240743, −2.70188081124891497875273702928, −1.91967007536020505093921080018, 0, 1.91967007536020505093921080018, 2.70188081124891497875273702928, 3.25314866125894232870360240743, 3.96635753409583245577902504697, 4.33299014941150381173882608243, 5.36508946029324143081225691311, 6.02549729779034229189676976711, 6.67003473767429922270918013273, 7.24249633231614401445733581277

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