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  + 2.23·2-s + 3-s + 2.98·4-s − 0.174·5-s + 2.23·6-s − 0.387·7-s + 2.19·8-s + 9-s − 0.390·10-s + 1.83·11-s + 2.98·12-s − 4.67·13-s − 0.865·14-s − 0.174·15-s − 1.06·16-s − 17-s + 2.23·18-s − 7.51·19-s − 0.521·20-s − 0.387·21-s + 4.08·22-s − 2.76·23-s + 2.19·24-s − 4.96·25-s − 10.4·26-s + 27-s − 1.15·28-s + ⋯
L(s)  = 1  + 1.57·2-s + 0.577·3-s + 1.49·4-s − 0.0781·5-s + 0.911·6-s − 0.146·7-s + 0.777·8-s + 0.333·9-s − 0.123·10-s + 0.551·11-s + 0.861·12-s − 1.29·13-s − 0.231·14-s − 0.0451·15-s − 0.265·16-s − 0.242·17-s + 0.526·18-s − 1.72·19-s − 0.116·20-s − 0.0845·21-s + 0.871·22-s − 0.576·23-s + 0.448·24-s − 0.993·25-s − 2.04·26-s + 0.192·27-s − 0.218·28-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 \( 1 - 2.23T + 2T^{2} \)
5 \( 1 + 0.174T + 5T^{2} \)
7 \( 1 + 0.387T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
19 \( 1 + 7.51T + 19T^{2} \)
23 \( 1 + 2.76T + 23T^{2} \)
29 \( 1 + 6.05T + 29T^{2} \)
31 \( 1 + 3.90T + 31T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 + 3.93T + 47T^{2} \)
53 \( 1 + 2.61T + 53T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 2.07T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 3.09T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 - 2.66T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.49521107945036036912816032298, −6.38953057199479194043107971034, −6.23468143262409403122742209244, −5.19454746801045816100726849092, −4.51728060359857056442259004270, −3.96667398055450242567232865544, −3.36778807579000560035399238550, −2.29398705252421099626262728876, −2.00043299793937172237342729140, 0, 2.00043299793937172237342729140, 2.29398705252421099626262728876, 3.36778807579000560035399238550, 3.96667398055450242567232865544, 4.51728060359857056442259004270, 5.19454746801045816100726849092, 6.23468143262409403122742209244, 6.38953057199479194043107971034, 7.49521107945036036912816032298

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