Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $
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 + 4-s + 5-s + 7-s + 8-s + 10-s + 13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 25-s + 26-s + 28-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 35-s + 2·37-s − 4·38-s + 40-s − 6·41-s + 8·43-s − 12·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8190\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.196701941$
$L(\frac12)$  $\approx$  $4.196701941$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−16.72575926041306, −16.50471148776070, −15.68209535835227, −14.97746634577308, −14.64224192852968, −14.03996387430738, −13.45258200373062, −12.87690762454356, −12.36031284324927, −11.60174232953872, −11.20816996331825, −10.30882456504039, −9.979199767226055, −9.151433974447080, −8.235225086409999, −7.909292853580616, −6.898273173821143, −6.414059613373089, −5.533586084686317, −5.213511402661488, −4.225820269170789, −3.618628549133593, −2.696503368450994, −1.909213152134956, −0.9441861801424986, 0.9441861801424986, 1.909213152134956, 2.696503368450994, 3.618628549133593, 4.225820269170789, 5.213511402661488, 5.533586084686317, 6.414059613373089, 6.898273173821143, 7.909292853580616, 8.235225086409999, 9.151433974447080, 9.979199767226055, 10.30882456504039, 11.20816996331825, 11.60174232953872, 12.36031284324927, 12.87690762454356, 13.45258200373062, 14.03996387430738, 14.64224192852968, 14.97746634577308, 15.68209535835227, 16.50471148776070, 16.72575926041306

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