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  + 0.0202·2-s − 3-s − 1.99·4-s − 1.35·5-s − 0.0202·6-s − 0.669·7-s − 0.0808·8-s + 9-s − 0.0274·10-s + 3.21·11-s + 1.99·12-s + 13-s − 0.0135·14-s + 1.35·15-s + 3.99·16-s − 0.412·17-s + 0.0202·18-s + 1.46·19-s + 2.71·20-s + 0.669·21-s + 0.0650·22-s − 2.51·23-s + 0.0808·24-s − 3.15·25-s + 0.0202·26-s − 27-s + 1.33·28-s + ⋯
L(s)  = 1  + 0.0142·2-s − 0.577·3-s − 0.999·4-s − 0.606·5-s − 0.00824·6-s − 0.252·7-s − 0.0285·8-s + 0.333·9-s − 0.00866·10-s + 0.970·11-s + 0.577·12-s + 0.277·13-s − 0.00361·14-s + 0.350·15-s + 0.999·16-s − 0.0999·17-s + 0.00476·18-s + 0.336·19-s + 0.606·20-s + 0.146·21-s + 0.0138·22-s − 0.523·23-s + 0.0164·24-s − 0.631·25-s + 0.00396·26-s − 0.192·27-s + 0.252·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7878716070$
$L(\frac12)$  $\approx$  $0.7878716070$
$L(\frac{3}{2})$   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 \( 1 - 0.0202T + 2T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 + 0.669T + 7T^{2} \)
11 \( 1 - 3.21T + 11T^{2} \)
17 \( 1 + 0.412T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 - 2.21T + 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + 9.85T + 41T^{2} \)
43 \( 1 + 7.39T + 43T^{2} \)
47 \( 1 + 0.629T + 47T^{2} \)
53 \( 1 - 6.18T + 53T^{2} \)
59 \( 1 + 0.139T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 9.27T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 2.85T + 73T^{2} \)
79 \( 1 + 3.10T + 79T^{2} \)
83 \( 1 - 7.16T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 2.47T + 97T^{2} \)
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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

−8.495426746052595443258281742386, −7.81684279582285424453872887003, −6.87679098821580767620644014626, −6.25239007041022624514497171590, −5.36191959592272208468988920115, −4.67365290122584835301596848686, −3.84807330861080340823706427411, −3.38363955611445202148934676571, −1.69610411542868069911210564028, −0.54090751412779429958864545248, 0.54090751412779429958864545248, 1.69610411542868069911210564028, 3.38363955611445202148934676571, 3.84807330861080340823706427411, 4.67365290122584835301596848686, 5.36191959592272208468988920115, 6.25239007041022624514497171590, 6.87679098821580767620644014626, 7.81684279582285424453872887003, 8.495426746052595443258281742386

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