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  + 0.0687·2-s − 3-s − 1.99·4-s + 0.966·5-s − 0.0687·6-s + 0.824·7-s − 0.274·8-s + 9-s + 0.0664·10-s + 2.54·11-s + 1.99·12-s − 0.576·13-s + 0.0567·14-s − 0.966·15-s + 3.97·16-s − 17-s + 0.0687·18-s + 3.08·19-s − 1.92·20-s − 0.824·21-s + 0.175·22-s − 2.04·23-s + 0.274·24-s − 4.06·25-s − 0.0396·26-s − 27-s − 1.64·28-s + ⋯
L(s)  = 1  + 0.0486·2-s − 0.577·3-s − 0.997·4-s + 0.432·5-s − 0.0280·6-s + 0.311·7-s − 0.0971·8-s + 0.333·9-s + 0.0210·10-s + 0.768·11-s + 0.575·12-s − 0.159·13-s + 0.0151·14-s − 0.249·15-s + 0.992·16-s − 0.242·17-s + 0.0162·18-s + 0.708·19-s − 0.431·20-s − 0.180·21-s + 0.0373·22-s − 0.426·23-s + 0.0560·24-s − 0.813·25-s − 0.00777·26-s − 0.192·27-s − 0.311·28-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 \( 1 - 0.0687T + 2T^{2} \)
5 \( 1 - 0.966T + 5T^{2} \)
7 \( 1 - 0.824T + 7T^{2} \)
11 \( 1 - 2.54T + 11T^{2} \)
13 \( 1 + 0.576T + 13T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 + 2.04T + 23T^{2} \)
29 \( 1 - 2.52T + 29T^{2} \)
31 \( 1 + 6.07T + 31T^{2} \)
37 \( 1 + 5.31T + 37T^{2} \)
41 \( 1 - 3.31T + 41T^{2} \)
43 \( 1 - 4.06T + 43T^{2} \)
47 \( 1 + 6.63T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 3.84T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 6.40T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 2.90T + 79T^{2} \)
83 \( 1 + 3.83T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−7.58276577009466200733117826851, −6.62422440723860624065516193809, −6.04833334283629732802790072572, −5.27676047266088618704141195522, −4.82517791158823751988677973671, −3.97868574549677702554057999204, −3.36133069107355080946185465272, −2.00979651830204386436476203770, −1.17258655137140479292782355871, 0, 1.17258655137140479292782355871, 2.00979651830204386436476203770, 3.36133069107355080946185465272, 3.97868574549677702554057999204, 4.82517791158823751988677973671, 5.27676047266088618704141195522, 6.04833334283629732802790072572, 6.62422440723860624065516193809, 7.58276577009466200733117826851

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