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  − 0.0648·2-s − 3-s − 1.99·4-s + 0.118·5-s + 0.0648·6-s − 4.97·7-s + 0.259·8-s + 9-s − 0.00769·10-s − 4.34·11-s + 1.99·12-s + 3.46·13-s + 0.322·14-s − 0.118·15-s + 3.97·16-s − 17-s − 0.0648·18-s − 0.806·19-s − 0.236·20-s + 4.97·21-s + 0.281·22-s + 5.73·23-s − 0.259·24-s − 4.98·25-s − 0.224·26-s − 27-s + 9.92·28-s + ⋯
L(s)  = 1  − 0.0458·2-s − 0.577·3-s − 0.997·4-s + 0.0530·5-s + 0.0264·6-s − 1.87·7-s + 0.0916·8-s + 0.333·9-s − 0.00243·10-s − 1.30·11-s + 0.576·12-s + 0.960·13-s + 0.0862·14-s − 0.0306·15-s + 0.993·16-s − 0.242·17-s − 0.0152·18-s − 0.185·19-s − 0.0529·20-s + 1.08·21-s + 0.0600·22-s + 1.19·23-s − 0.0529·24-s − 0.997·25-s − 0.0440·26-s − 0.192·27-s + 1.87·28-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 \( 1 + 0.0648T + 2T^{2} \)
5 \( 1 - 0.118T + 5T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 + 4.34T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
19 \( 1 + 0.806T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 + 6.51T + 29T^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + 2.01T + 37T^{2} \)
41 \( 1 - 6.33T + 41T^{2} \)
43 \( 1 - 3.21T + 43T^{2} \)
47 \( 1 - 9.60T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 2.31T + 59T^{2} \)
61 \( 1 + 7.66T + 61T^{2} \)
67 \( 1 - 8.05T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + 8.31T + 97T^{2} \)
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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

−7.47226091996315292978661422647, −6.71847497421561006306707364481, −5.87322288463603780077606694764, −5.62551167176792435559840539443, −4.71992654339974940213200008428, −3.78098056579282790596196353251, −3.36605407930863125148048618794, −2.31494571444002562185816633792, −0.798614337122434857747102607499, 0, 0.798614337122434857747102607499, 2.31494571444002562185816633792, 3.36605407930863125148048618794, 3.78098056579282790596196353251, 4.71992654339974940213200008428, 5.62551167176792435559840539443, 5.87322288463603780077606694764, 6.71847497421561006306707364481, 7.47226091996315292978661422647

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