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  + 2.73·2-s − 3-s + 5.49·4-s + 2.25·5-s − 2.73·6-s − 2.23·7-s + 9.57·8-s + 9-s + 6.16·10-s + 4.01·11-s − 5.49·12-s + 13-s − 6.10·14-s − 2.25·15-s + 15.2·16-s + 1.93·17-s + 2.73·18-s − 3.89·19-s + 12.3·20-s + 2.23·21-s + 10.9·22-s + 1.46·23-s − 9.57·24-s + 0.0720·25-s + 2.73·26-s − 27-s − 12.2·28-s + ⋯
L(s)  = 1  + 1.93·2-s − 0.577·3-s + 2.74·4-s + 1.00·5-s − 1.11·6-s − 0.843·7-s + 3.38·8-s + 0.333·9-s + 1.94·10-s + 1.20·11-s − 1.58·12-s + 0.277·13-s − 1.63·14-s − 0.581·15-s + 3.80·16-s + 0.469·17-s + 0.645·18-s − 0.894·19-s + 2.76·20-s + 0.486·21-s + 2.34·22-s + 0.305·23-s − 1.95·24-s + 0.0144·25-s + 0.536·26-s − 0.192·27-s − 2.31·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.174447620$
$L(\frac12)$  $\approx$  $7.174447620$
$L(\frac{3}{2})$   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 \( 1 - 2.73T + 2T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
11 \( 1 - 4.01T + 11T^{2} \)
17 \( 1 - 1.93T + 17T^{2} \)
19 \( 1 + 3.89T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 0.0392T + 29T^{2} \)
31 \( 1 + 0.286T + 31T^{2} \)
37 \( 1 - 0.862T + 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 - 3.27T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 1.47T + 61T^{2} \)
67 \( 1 - 3.33T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 - 7.69T + 73T^{2} \)
79 \( 1 - 3.50T + 79T^{2} \)
83 \( 1 + 6.37T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−8.264256669930640147748619666828, −6.97003926931721826038780740704, −6.69586834247377137393240763627, −5.98825027394617778287579782974, −5.63167677146058602251446226566, −4.69879387574029760324259917345, −3.93313917599979214203188385670, −3.24957517608113811660693399076, −2.21923593343642522226343299142, −1.35426036152705946998885207700, 1.35426036152705946998885207700, 2.21923593343642522226343299142, 3.24957517608113811660693399076, 3.93313917599979214203188385670, 4.69879387574029760324259917345, 5.63167677146058602251446226566, 5.98825027394617778287579782974, 6.69586834247377137393240763627, 6.97003926931721826038780740704, 8.264256669930640147748619666828

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