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  − 1.38·2-s − 3-s − 0.0885·4-s + 2.12·5-s + 1.38·6-s − 0.203·7-s + 2.88·8-s + 9-s − 2.94·10-s − 1.88·11-s + 0.0885·12-s + 3.56·13-s + 0.281·14-s − 2.12·15-s − 3.81·16-s − 17-s − 1.38·18-s − 0.0124·19-s − 0.188·20-s + 0.203·21-s + 2.60·22-s + 6.58·23-s − 2.88·24-s − 0.474·25-s − 4.92·26-s − 27-s + 0.0180·28-s + ⋯
L(s)  = 1  − 0.977·2-s − 0.577·3-s − 0.0442·4-s + 0.951·5-s + 0.564·6-s − 0.0770·7-s + 1.02·8-s + 0.333·9-s − 0.930·10-s − 0.567·11-s + 0.0255·12-s + 0.988·13-s + 0.0753·14-s − 0.549·15-s − 0.953·16-s − 0.242·17-s − 0.325·18-s − 0.00285·19-s − 0.0421·20-s + 0.0445·21-s + 0.554·22-s + 1.37·23-s − 0.589·24-s − 0.0949·25-s − 0.966·26-s − 0.192·27-s + 0.00341·28-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 \( 1 + 1.38T + 2T^{2} \)
5 \( 1 - 2.12T + 5T^{2} \)
7 \( 1 + 0.203T + 7T^{2} \)
11 \( 1 + 1.88T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
19 \( 1 + 0.0124T + 19T^{2} \)
23 \( 1 - 6.58T + 23T^{2} \)
29 \( 1 - 0.647T + 29T^{2} \)
31 \( 1 - 0.398T + 31T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 - 5.83T + 47T^{2} \)
53 \( 1 + 1.05T + 53T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 - 2.57T + 61T^{2} \)
67 \( 1 - 6.56T + 67T^{2} \)
71 \( 1 - 3.69T + 71T^{2} \)
73 \( 1 - 0.453T + 73T^{2} \)
79 \( 1 + 2.56T + 79T^{2} \)
83 \( 1 - 8.30T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−7.61213374044503571159433720619, −6.72456859312871463530861832028, −6.30513885119871287464516041503, −5.30285609899960454169427781793, −4.96873725462056531094340324339, −3.96009890727752934381373248827, −2.93709976748810573873609819724, −1.80684749592890611441395034733, −1.17387196158058177258123325999, 0, 1.17387196158058177258123325999, 1.80684749592890611441395034733, 2.93709976748810573873609819724, 3.96009890727752934381373248827, 4.96873725462056531094340324339, 5.30285609899960454169427781793, 6.30513885119871287464516041503, 6.72456859312871463530861832028, 7.61213374044503571159433720619

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