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  − 0.503·2-s + 3-s − 1.74·4-s − 1.55·5-s − 0.503·6-s + 2.07·7-s + 1.88·8-s + 9-s + 0.783·10-s − 2.88·11-s − 1.74·12-s + 2.62·13-s − 1.04·14-s − 1.55·15-s + 2.54·16-s − 17-s − 0.503·18-s + 5.60·19-s + 2.72·20-s + 2.07·21-s + 1.44·22-s + 0.593·23-s + 1.88·24-s − 2.57·25-s − 1.31·26-s + 27-s − 3.62·28-s + ⋯
L(s)  = 1  − 0.355·2-s + 0.577·3-s − 0.873·4-s − 0.696·5-s − 0.205·6-s + 0.785·7-s + 0.666·8-s + 0.333·9-s + 0.247·10-s − 0.868·11-s − 0.504·12-s + 0.727·13-s − 0.279·14-s − 0.402·15-s + 0.636·16-s − 0.242·17-s − 0.118·18-s + 1.28·19-s + 0.608·20-s + 0.453·21-s + 0.308·22-s + 0.123·23-s + 0.384·24-s − 0.514·25-s − 0.258·26-s + 0.192·27-s − 0.685·28-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 \( 1 + 0.503T + 2T^{2} \)
5 \( 1 + 1.55T + 5T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 - 0.593T + 23T^{2} \)
29 \( 1 + 9.84T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + 8.19T + 41T^{2} \)
43 \( 1 + 4.66T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 7.63T + 59T^{2} \)
61 \( 1 - 3.42T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 2.18T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 - 7.90T + 89T^{2} \)
97 \( 1 - 7.71T + 97T^{2} \)
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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

−7.55213315981107930492658938313, −7.34512484586069749595120230285, −6.00354290677382559339961154625, −5.15332659224334821590754668293, −4.73097280594119447589309087495, −3.72338254469726087311465783267, −3.39736547942016766622419192934, −2.08890247574035201336196157088, −1.19647127682028224143356119971, 0, 1.19647127682028224143356119971, 2.08890247574035201336196157088, 3.39736547942016766622419192934, 3.72338254469726087311465783267, 4.73097280594119447589309087495, 5.15332659224334821590754668293, 6.00354290677382559339961154625, 7.34512484586069749595120230285, 7.55213315981107930492658938313

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