Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.0371·2-s − 3-s − 1.99·4-s + 2.55·5-s + 0.0371·6-s + 1.94·7-s + 0.148·8-s + 9-s − 0.0946·10-s + 0.0498·11-s + 1.99·12-s + 13-s − 0.0722·14-s − 2.55·15-s + 3.99·16-s − 4.77·17-s − 0.0371·18-s − 6.56·19-s − 5.09·20-s − 1.94·21-s − 0.00185·22-s + 5.81·23-s − 0.148·24-s + 1.50·25-s − 0.0371·26-s − 27-s − 3.89·28-s + ⋯
L(s)  = 1  − 0.0262·2-s − 0.577·3-s − 0.999·4-s + 1.14·5-s + 0.0151·6-s + 0.735·7-s + 0.0524·8-s + 0.333·9-s − 0.0299·10-s + 0.0150·11-s + 0.576·12-s + 0.277·13-s − 0.0193·14-s − 0.658·15-s + 0.997·16-s − 1.15·17-s − 0.00875·18-s − 1.50·19-s − 1.13·20-s − 0.424·21-s − 0.000394·22-s + 1.21·23-s − 0.0303·24-s + 0.300·25-s − 0.00728·26-s − 0.192·27-s − 0.735·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.596278819$
$L(\frac12)$  $\approx$  $1.596278819$
$L(\frac{3}{2})$   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 \( 1 + 0.0371T + 2T^{2} \)
5 \( 1 - 2.55T + 5T^{2} \)
7 \( 1 - 1.94T + 7T^{2} \)
11 \( 1 - 0.0498T + 11T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 6.56T + 19T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 - 7.77T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 + 4.46T + 41T^{2} \)
43 \( 1 + 0.323T + 43T^{2} \)
47 \( 1 + 2.86T + 47T^{2} \)
53 \( 1 - 2.38T + 53T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.542521187729396930090521930428, −7.954058193799495392467872360355, −6.54522020792792144630677417120, −6.38241394944303416872415261969, −5.33963829476489216708516589577, −4.67980407549216604675696889298, −4.29061354905136301406221868994, −2.83546440679675573318705916579, −1.80587030658753093164538340839, −0.78406440423641847536723469655, 0.78406440423641847536723469655, 1.80587030658753093164538340839, 2.83546440679675573318705916579, 4.29061354905136301406221868994, 4.67980407549216604675696889298, 5.33963829476489216708516589577, 6.38241394944303416872415261969, 6.54522020792792144630677417120, 7.954058193799495392467872360355, 8.542521187729396930090521930428

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