Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.491·2-s − 3-s − 1.75·4-s − 1.95·5-s + 0.491·6-s − 2.76·7-s + 1.84·8-s + 9-s + 0.961·10-s + 0.480·11-s + 1.75·12-s + 13-s + 1.36·14-s + 1.95·15-s + 2.60·16-s − 2.47·17-s − 0.491·18-s − 0.387·19-s + 3.44·20-s + 2.76·21-s − 0.236·22-s − 2.25·23-s − 1.84·24-s − 1.17·25-s − 0.491·26-s − 27-s + 4.86·28-s + ⋯
L(s)  = 1  − 0.347·2-s − 0.577·3-s − 0.879·4-s − 0.875·5-s + 0.200·6-s − 1.04·7-s + 0.653·8-s + 0.333·9-s + 0.304·10-s + 0.144·11-s + 0.507·12-s + 0.277·13-s + 0.363·14-s + 0.505·15-s + 0.652·16-s − 0.601·17-s − 0.115·18-s − 0.0888·19-s + 0.769·20-s + 0.603·21-s − 0.0503·22-s − 0.469·23-s − 0.377·24-s − 0.234·25-s − 0.0963·26-s − 0.192·27-s + 0.919·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1559874685$
$L(\frac12)$  $\approx$  $0.1559874685$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 + 0.491T + 2T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 - 0.480T + 11T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + 0.387T + 19T^{2} \)
23 \( 1 + 2.25T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 + 8.05T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 0.735T + 43T^{2} \)
47 \( 1 - 6.95T + 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 5.00T + 61T^{2} \)
67 \( 1 + 6.94T + 67T^{2} \)
71 \( 1 + 6.52T + 71T^{2} \)
73 \( 1 + 4.41T + 73T^{2} \)
79 \( 1 - 1.22T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 - 8.28T + 97T^{2} \)
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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

−8.560760542928014420765536153788, −7.56210985816624560725080885366, −7.25550206060773410282378921181, −6.10087200958183584415941350581, −5.61819570195801671865875531911, −4.46599959780870948463757895285, −3.97608499220409889647076533854, −3.23942652308250744697275994430, −1.68247628052782185279641780134, −0.24581619256385634309094619643, 0.24581619256385634309094619643, 1.68247628052782185279641780134, 3.23942652308250744697275994430, 3.97608499220409889647076533854, 4.46599959780870948463757895285, 5.61819570195801671865875531911, 6.10087200958183584415941350581, 7.25550206060773410282378921181, 7.56210985816624560725080885366, 8.560760542928014420765536153788

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