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  + 0.422·2-s + 3-s − 1.82·4-s − 0.274·5-s + 0.422·6-s − 5.26·7-s − 1.61·8-s + 9-s − 0.116·10-s + 2.24·11-s − 1.82·12-s + 2.71·13-s − 2.22·14-s − 0.274·15-s + 2.95·16-s − 17-s + 0.422·18-s + 3.58·19-s + 0.500·20-s − 5.26·21-s + 0.950·22-s − 4.99·23-s − 1.61·24-s − 4.92·25-s + 1.14·26-s + 27-s + 9.59·28-s + ⋯
L(s)  = 1  + 0.298·2-s + 0.577·3-s − 0.910·4-s − 0.122·5-s + 0.172·6-s − 1.99·7-s − 0.571·8-s + 0.333·9-s − 0.0367·10-s + 0.677·11-s − 0.525·12-s + 0.753·13-s − 0.595·14-s − 0.0709·15-s + 0.739·16-s − 0.242·17-s + 0.0996·18-s + 0.822·19-s + 0.111·20-s − 1.14·21-s + 0.202·22-s − 1.04·23-s − 0.329·24-s − 0.984·25-s + 0.225·26-s + 0.192·27-s + 1.81·28-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 \( 1 - 0.422T + 2T^{2} \)
5 \( 1 + 0.274T + 5T^{2} \)
7 \( 1 + 5.26T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 - 2.71T + 13T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 + 4.99T + 23T^{2} \)
29 \( 1 + 3.94T + 29T^{2} \)
31 \( 1 - 7.05T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 4.05T + 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 + 7.42T + 59T^{2} \)
61 \( 1 - 1.49T + 61T^{2} \)
67 \( 1 - 0.290T + 67T^{2} \)
71 \( 1 - 7.85T + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 - 2.79T + 79T^{2} \)
83 \( 1 + 2.71T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−7.61150525661135012210966047923, −6.56354653174500197150006507448, −6.17178968561465587924284996268, −5.53566186313727965884325391939, −4.27567635777198368565624244389, −3.89345771803350739315918942347, −3.29856831527203427261859591190, −2.56643076579240850155231938475, −1.11116432799895147395346002334, 0, 1.11116432799895147395346002334, 2.56643076579240850155231938475, 3.29856831527203427261859591190, 3.89345771803350739315918942347, 4.27567635777198368565624244389, 5.53566186313727965884325391939, 6.17178968561465587924284996268, 6.56354653174500197150006507448, 7.61150525661135012210966047923

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