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  + 0.940·2-s − 3-s − 1.11·4-s − 1.95·5-s − 0.940·6-s − 0.363·7-s − 2.92·8-s + 9-s − 1.83·10-s + 1.99·11-s + 1.11·12-s + 13-s − 0.341·14-s + 1.95·15-s − 0.521·16-s − 2.22·17-s + 0.940·18-s − 5.96·19-s + 2.17·20-s + 0.363·21-s + 1.87·22-s − 1.97·23-s + 2.92·24-s − 1.19·25-s + 0.940·26-s − 27-s + 0.405·28-s + ⋯
L(s)  = 1  + 0.664·2-s − 0.577·3-s − 0.558·4-s − 0.872·5-s − 0.383·6-s − 0.137·7-s − 1.03·8-s + 0.333·9-s − 0.579·10-s + 0.602·11-s + 0.322·12-s + 0.277·13-s − 0.0912·14-s + 0.503·15-s − 0.130·16-s − 0.540·17-s + 0.221·18-s − 1.36·19-s + 0.486·20-s + 0.0792·21-s + 0.400·22-s − 0.412·23-s + 0.597·24-s − 0.238·25-s + 0.184·26-s − 0.192·27-s + 0.0766·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8005955759$
$L(\frac12)$  $\approx$  $0.8005955759$
$L(\frac{3}{2})$   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 \( 1 - 0.940T + 2T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
7 \( 1 + 0.363T + 7T^{2} \)
11 \( 1 - 1.99T + 11T^{2} \)
17 \( 1 + 2.22T + 17T^{2} \)
19 \( 1 + 5.96T + 19T^{2} \)
23 \( 1 + 1.97T + 23T^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
31 \( 1 + 6.18T + 31T^{2} \)
37 \( 1 - 2.13T + 37T^{2} \)
41 \( 1 + 0.409T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 1.45T + 53T^{2} \)
59 \( 1 + 2.75T + 59T^{2} \)
61 \( 1 + 2.43T + 61T^{2} \)
67 \( 1 - 0.0721T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 8.03T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 8.54T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.371225919215319820338240989675, −7.77374488034671271826916673456, −6.67426227984074839893874201629, −6.23088387109360969376775883237, −5.38144382192121143317967224229, −4.50461836214089917012519051831, −4.01891579154996419992822372886, −3.39725314425908771579295444503, −2.01495372582180235450289882683, −0.46471390766637537831850624826, 0.46471390766637537831850624826, 2.01495372582180235450289882683, 3.39725314425908771579295444503, 4.01891579154996419992822372886, 4.50461836214089917012519051831, 5.38144382192121143317967224229, 6.23088387109360969376775883237, 6.67426227984074839893874201629, 7.77374488034671271826916673456, 8.371225919215319820338240989675

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