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  + 0.515·2-s + 3-s − 1.73·4-s − 1.23·5-s + 0.515·6-s + 1.08·7-s − 1.92·8-s + 9-s − 0.635·10-s + 1.20·11-s − 1.73·12-s − 2.47·13-s + 0.560·14-s − 1.23·15-s + 2.47·16-s − 17-s + 0.515·18-s + 1.19·19-s + 2.13·20-s + 1.08·21-s + 0.619·22-s + 1.29·23-s − 1.92·24-s − 3.48·25-s − 1.27·26-s + 27-s − 1.88·28-s + ⋯
L(s)  = 1  + 0.364·2-s + 0.577·3-s − 0.866·4-s − 0.551·5-s + 0.210·6-s + 0.410·7-s − 0.681·8-s + 0.333·9-s − 0.201·10-s + 0.361·11-s − 0.500·12-s − 0.685·13-s + 0.149·14-s − 0.318·15-s + 0.618·16-s − 0.242·17-s + 0.121·18-s + 0.273·19-s + 0.477·20-s + 0.236·21-s + 0.131·22-s + 0.269·23-s − 0.393·24-s − 0.696·25-s − 0.249·26-s + 0.192·27-s − 0.355·28-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 \( 1 - 0.515T + 2T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 1.08T + 7T^{2} \)
11 \( 1 - 1.20T + 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
19 \( 1 - 1.19T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 - 4.08T + 29T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 - 0.0125T + 37T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 + 0.927T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 3.78T + 53T^{2} \)
59 \( 1 - 8.12T + 59T^{2} \)
61 \( 1 + 6.19T + 61T^{2} \)
67 \( 1 + 5.60T + 67T^{2} \)
71 \( 1 - 5.68T + 71T^{2} \)
73 \( 1 - 1.77T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + 9.02T + 83T^{2} \)
89 \( 1 + 6.10T + 89T^{2} \)
97 \( 1 - 8.12T + 97T^{2} \)
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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

−7.64646929781853211987969024841, −6.88049685439031820400927718291, −6.04544033109275392488365964742, −5.09858895638642307218918794333, −4.64689170120150841057033077646, −3.91338768484550554876990688059, −3.30727561951195263061534929809, −2.40296700539607350664471335704, −1.24887529611652337040486696325, 0, 1.24887529611652337040486696325, 2.40296700539607350664471335704, 3.30727561951195263061534929809, 3.91338768484550554876990688059, 4.64689170120150841057033077646, 5.09858895638642307218918794333, 6.04544033109275392488365964742, 6.88049685439031820400927718291, 7.64646929781853211987969024841

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