Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82·2-s − 3-s + 1.33·4-s + 0.0479·5-s + 1.82·6-s − 0.416·7-s + 1.21·8-s + 9-s − 0.0876·10-s + 2.02·11-s − 1.33·12-s − 4.18·13-s + 0.761·14-s − 0.0479·15-s − 4.88·16-s + 17-s − 1.82·18-s − 3.54·19-s + 0.0640·20-s + 0.416·21-s − 3.68·22-s − 4.19·23-s − 1.21·24-s − 4.99·25-s + 7.64·26-s − 27-s − 0.556·28-s + ⋯
L(s)  = 1  − 1.29·2-s − 0.577·3-s + 0.667·4-s + 0.0214·5-s + 0.745·6-s − 0.157·7-s + 0.428·8-s + 0.333·9-s − 0.0277·10-s + 0.609·11-s − 0.385·12-s − 1.16·13-s + 0.203·14-s − 0.0123·15-s − 1.22·16-s + 0.242·17-s − 0.430·18-s − 0.813·19-s + 0.0143·20-s + 0.0909·21-s − 0.786·22-s − 0.875·23-s − 0.247·24-s − 0.999·25-s + 1.49·26-s − 0.192·27-s − 0.105·28-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  =  0
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2731291503$
$L(\frac12)$  $\approx$  $0.2731291503$
$L(\frac{3}{2})$   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 \( 1 + 1.82T + 2T^{2} \)
5 \( 1 - 0.0479T + 5T^{2} \)
7 \( 1 + 0.416T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 4.18T + 13T^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 + 4.19T + 23T^{2} \)
29 \( 1 + 0.335T + 29T^{2} \)
31 \( 1 + 8.59T + 31T^{2} \)
37 \( 1 - 6.45T + 37T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 8.67T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 9.03T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 - 2.46T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 + 3.38T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.980863236051888223077092451341, −7.19627683631495267238209721856, −6.70536438471146216180501248001, −5.92721389140914434389131114994, −5.06976386253629181257227196664, −4.35074864939430520435546064596, −3.53981018588727803455381096396, −2.18062967294961392015343480125, −1.61938900759532371269300642423, −0.32459617194791684929793950288, 0.32459617194791684929793950288, 1.61938900759532371269300642423, 2.18062967294961392015343480125, 3.53981018588727803455381096396, 4.35074864939430520435546064596, 5.06976386253629181257227196664, 5.92721389140914434389131114994, 6.70536438471146216180501248001, 7.19627683631495267238209721856, 7.980863236051888223077092451341

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