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  + 2.28·2-s − 3-s + 3.22·4-s − 1.71·5-s − 2.28·6-s − 3.41·7-s + 2.79·8-s + 9-s − 3.91·10-s − 2.39·11-s − 3.22·12-s + 13-s − 7.80·14-s + 1.71·15-s − 0.0631·16-s + 0.255·17-s + 2.28·18-s + 5.27·19-s − 5.52·20-s + 3.41·21-s − 5.46·22-s + 8.19·23-s − 2.79·24-s − 2.05·25-s + 2.28·26-s − 27-s − 10.9·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 0.577·3-s + 1.61·4-s − 0.766·5-s − 0.932·6-s − 1.29·7-s + 0.987·8-s + 0.333·9-s − 1.23·10-s − 0.721·11-s − 0.930·12-s + 0.277·13-s − 2.08·14-s + 0.442·15-s − 0.0157·16-s + 0.0618·17-s + 0.538·18-s + 1.21·19-s − 1.23·20-s + 0.744·21-s − 1.16·22-s + 1.70·23-s − 0.569·24-s − 0.411·25-s + 0.448·26-s − 0.192·27-s − 2.07·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.670574142$
$L(\frac12)$  $\approx$  $2.670574142$
$L(\frac{3}{2})$   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 \( 1 - 2.28T + 2T^{2} \)
5 \( 1 + 1.71T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 2.39T + 11T^{2} \)
17 \( 1 - 0.255T + 17T^{2} \)
19 \( 1 - 5.27T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 4.00T + 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 - 2.94T + 41T^{2} \)
43 \( 1 - 3.05T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 0.884T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 7.71T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 8.26T + 73T^{2} \)
79 \( 1 - 5.00T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−8.189407446173026556461341411659, −7.30195661240676580359544906185, −6.76506491512873614124354810443, −6.08063905109876231700176171432, −5.33010074381035990773266423248, −4.78058799596497526331595976328, −3.80012441605842221784048607680, −3.27861492363998906185263110596, −2.53980358706423237577926736188, −0.73359261335936170791267302204, 0.73359261335936170791267302204, 2.53980358706423237577926736188, 3.27861492363998906185263110596, 3.80012441605842221784048607680, 4.78058799596497526331595976328, 5.33010074381035990773266423248, 6.08063905109876231700176171432, 6.76506491512873614124354810443, 7.30195661240676580359544906185, 8.189407446173026556461341411659

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