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  − 2.00·2-s − 3-s + 2.02·4-s − 1.83·5-s + 2.00·6-s − 1.92·7-s − 0.0524·8-s + 9-s + 3.67·10-s + 3.86·11-s − 2.02·12-s + 6.14·13-s + 3.86·14-s + 1.83·15-s − 3.94·16-s − 17-s − 2.00·18-s + 0.890·19-s − 3.71·20-s + 1.92·21-s − 7.75·22-s − 5.99·23-s + 0.0524·24-s − 1.64·25-s − 12.3·26-s − 27-s − 3.90·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1.01·4-s − 0.819·5-s + 0.819·6-s − 0.727·7-s − 0.0185·8-s + 0.333·9-s + 1.16·10-s + 1.16·11-s − 0.584·12-s + 1.70·13-s + 1.03·14-s + 0.473·15-s − 0.986·16-s − 0.242·17-s − 0.472·18-s + 0.204·19-s − 0.830·20-s + 0.420·21-s − 1.65·22-s − 1.25·23-s + 0.0107·24-s − 0.328·25-s − 2.41·26-s − 0.192·27-s − 0.737·28-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 \( 1 + 2.00T + 2T^{2} \)
5 \( 1 + 1.83T + 5T^{2} \)
7 \( 1 + 1.92T + 7T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 - 6.14T + 13T^{2} \)
19 \( 1 - 0.890T + 19T^{2} \)
23 \( 1 + 5.99T + 23T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + 6.65T + 31T^{2} \)
37 \( 1 - 3.43T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 9.99T + 43T^{2} \)
47 \( 1 + 1.34T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 2.14T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 4.33T + 67T^{2} \)
71 \( 1 - 8.04T + 71T^{2} \)
73 \( 1 + 0.195T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 9.85T + 83T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
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

−7.68561735068690260428600343631, −6.89997831518102059434891266615, −6.24633874108527109458005543597, −5.85743449358165123270685910269, −4.26861933423987483761005147880, −4.10514307028295669669618297393, −3.07208788348793398647520028466, −1.70692185122469777339576665517, −0.970126595997107059479072529798, 0, 0.970126595997107059479072529798, 1.70692185122469777339576665517, 3.07208788348793398647520028466, 4.10514307028295669669618297393, 4.26861933423987483761005147880, 5.85743449358165123270685910269, 6.24633874108527109458005543597, 6.89997831518102059434891266615, 7.68561735068690260428600343631

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