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  − 1.95·2-s − 3-s + 1.81·4-s − 1.54·5-s + 1.95·6-s + 2.71·7-s + 0.369·8-s + 9-s + 3.01·10-s − 1.66·11-s − 1.81·12-s − 3.33·13-s − 5.29·14-s + 1.54·15-s − 4.34·16-s − 17-s − 1.95·18-s − 1.70·19-s − 2.79·20-s − 2.71·21-s + 3.24·22-s + 4.45·23-s − 0.369·24-s − 2.61·25-s + 6.50·26-s − 27-s + 4.91·28-s + ⋯
L(s)  = 1  − 1.38·2-s − 0.577·3-s + 0.905·4-s − 0.691·5-s + 0.796·6-s + 1.02·7-s + 0.130·8-s + 0.333·9-s + 0.954·10-s − 0.501·11-s − 0.522·12-s − 0.924·13-s − 1.41·14-s + 0.399·15-s − 1.08·16-s − 0.242·17-s − 0.460·18-s − 0.391·19-s − 0.625·20-s − 0.592·21-s + 0.692·22-s + 0.928·23-s − 0.0753·24-s − 0.522·25-s + 1.27·26-s − 0.192·27-s + 0.928·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

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 \( 1 + 1.95T + 2T^{2} \)
5 \( 1 + 1.54T + 5T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
19 \( 1 + 1.70T + 19T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 - 3.74T + 29T^{2} \)
31 \( 1 - 1.92T + 31T^{2} \)
37 \( 1 + 5.87T + 37T^{2} \)
41 \( 1 - 9.83T + 41T^{2} \)
43 \( 1 - 9.63T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 6.79T + 53T^{2} \)
59 \( 1 + 2.46T + 59T^{2} \)
61 \( 1 + 2.56T + 61T^{2} \)
67 \( 1 + 0.665T + 67T^{2} \)
71 \( 1 - 7.81T + 71T^{2} \)
73 \( 1 + 0.675T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 6.98T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 5.15T + 97T^{2} \)
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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

−7.53303788608162627455765992114, −7.23256137399156536414272196967, −6.34992212838996481972704925514, −5.30765093467786843882424359126, −4.69250923060694742172013350512, −4.14822629827437843536274342204, −2.76350508589388000869138042956, −1.91149586353218468849855200902, −0.932199538729390722811300212816, 0, 0.932199538729390722811300212816, 1.91149586353218468849855200902, 2.76350508589388000869138042956, 4.14822629827437843536274342204, 4.69250923060694742172013350512, 5.30765093467786843882424359126, 6.34992212838996481972704925514, 7.23256137399156536414272196967, 7.53303788608162627455765992114

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