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  − 2.27·2-s − 3-s + 3.17·4-s + 2.61·5-s + 2.27·6-s + 3.54·7-s − 2.67·8-s + 9-s − 5.94·10-s + 2.60·11-s − 3.17·12-s − 6.26·13-s − 8.06·14-s − 2.61·15-s − 0.263·16-s − 17-s − 2.27·18-s + 2.01·19-s + 8.30·20-s − 3.54·21-s − 5.93·22-s − 8.87·23-s + 2.67·24-s + 1.82·25-s + 14.2·26-s − 27-s + 11.2·28-s + ⋯
L(s)  = 1  − 1.60·2-s − 0.577·3-s + 1.58·4-s + 1.16·5-s + 0.928·6-s + 1.34·7-s − 0.946·8-s + 0.333·9-s − 1.88·10-s + 0.785·11-s − 0.916·12-s − 1.73·13-s − 2.15·14-s − 0.674·15-s − 0.0657·16-s − 0.242·17-s − 0.536·18-s + 0.461·19-s + 1.85·20-s − 0.773·21-s − 1.26·22-s − 1.85·23-s + 0.546·24-s + 0.365·25-s + 2.79·26-s − 0.192·27-s + 2.12·28-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 \( 1 + 2.27T + 2T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 + 6.26T + 13T^{2} \)
19 \( 1 - 2.01T + 19T^{2} \)
23 \( 1 + 8.87T + 23T^{2} \)
29 \( 1 + 5.96T + 29T^{2} \)
31 \( 1 - 1.87T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 - 6.75T + 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 + 7.43T + 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 - 8.10T + 59T^{2} \)
61 \( 1 - 9.35T + 61T^{2} \)
67 \( 1 + 0.624T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 - 17.5T + 83T^{2} \)
89 \( 1 + 0.0410T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−7.82203993631767433861629149903, −6.89551330416359410914892861245, −6.36522550674216065267400666155, −5.45830321006543767856359666062, −4.92585760151833895898602010121, −4.02242564146897203196922542049, −2.35171393590025816448743440821, −1.95569853163825849238446525213, −1.24887733181702100312631665669, 0, 1.24887733181702100312631665669, 1.95569853163825849238446525213, 2.35171393590025816448743440821, 4.02242564146897203196922542049, 4.92585760151833895898602010121, 5.45830321006543767856359666062, 6.36522550674216065267400666155, 6.89551330416359410914892861245, 7.82203993631767433861629149903

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