Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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-s − 0.635·3-s + 4-s − 5-s − 0.635·6-s + 1.23·7-s + 8-s − 2.59·9-s − 10-s + 0.272·11-s − 0.635·12-s − 13-s + 1.23·14-s + 0.635·15-s + 16-s + 7.11·17-s − 2.59·18-s − 1.95·19-s − 20-s − 0.782·21-s + 0.272·22-s − 1.59·23-s − 0.635·24-s + 25-s − 26-s + 3.55·27-s + 1.23·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.366·3-s + 0.5·4-s − 0.447·5-s − 0.259·6-s + 0.465·7-s + 0.353·8-s − 0.865·9-s − 0.316·10-s + 0.0821·11-s − 0.183·12-s − 0.277·13-s + 0.329·14-s + 0.163·15-s + 0.250·16-s + 1.72·17-s − 0.612·18-s − 0.448·19-s − 0.223·20-s − 0.170·21-s + 0.0581·22-s − 0.332·23-s − 0.129·24-s + 0.200·25-s − 0.196·26-s + 0.683·27-s + 0.232·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.377768020$
$L(\frac12)$  $\approx$  $2.377768020$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 + T \)
good3 \( 1 + 0.635T + 3T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 0.272T + 11T^{2} \)
17 \( 1 - 7.11T + 17T^{2} \)
19 \( 1 + 1.95T + 19T^{2} \)
23 \( 1 + 1.59T + 23T^{2} \)
29 \( 1 - 0.671T + 29T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 - 2.10T + 47T^{2} \)
53 \( 1 - 7.92T + 53T^{2} \)
59 \( 1 - 8.12T + 59T^{2} \)
61 \( 1 + 8.58T + 61T^{2} \)
67 \( 1 - 4.35T + 67T^{2} \)
71 \( 1 - 5.14T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 9.00T + 83T^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.146745013155227647957647523388, −7.80010244610485576608621376468, −6.85639671509514109318893212247, −6.06206437390508988622062200788, −5.39796827067690353737975264871, −4.81279720637462332717009042624, −3.83861942618008255810920856677, −3.13346303150528607789275852023, −2.13345603621137107650233993390, −0.808882383756333773341717806240, 0.808882383756333773341717806240, 2.13345603621137107650233993390, 3.13346303150528607789275852023, 3.83861942618008255810920856677, 4.81279720637462332717009042624, 5.39796827067690353737975264871, 6.06206437390508988622062200788, 6.85639671509514109318893212247, 7.80010244610485576608621376468, 8.146745013155227647957647523388

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