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 − 4·31-s − 32-s + 6·34-s − 35-s − 10·37-s + 4·38-s + 40-s − 6·41-s + 8·43-s + 49-s − 50-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 − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.141·50-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  =  $(2,\ 8190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8709071618$
$L(\frac12)$  $\approx$  $0.8709071618$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−17.08200938497298, −16.32896588729640, −15.90034952181462, −15.16115637781206, −14.92143352907132, −14.09858637624693, −13.32179582574864, −12.84839908282130, −12.06543392431495, −11.49220187927445, −10.86197084415405, −10.59353534859256, −9.675522879887256, −8.891041779408560, −8.605157369305601, −7.938641635477664, −7.036303219360571, −6.799695981959259, −5.796064524277567, −5.082426662276095, −4.128700662051059, −3.588936564328635, −2.353179975037027, −1.812566102808810, −0.4964826358254509, 0.4964826358254509, 1.812566102808810, 2.353179975037027, 3.588936564328635, 4.128700662051059, 5.082426662276095, 5.796064524277567, 6.799695981959259, 7.036303219360571, 7.938641635477664, 8.605157369305601, 8.891041779408560, 9.675522879887256, 10.59353534859256, 10.86197084415405, 11.49220187927445, 12.06543392431495, 12.84839908282130, 13.32179582574864, 14.09858637624693, 14.92143352907132, 15.16115637781206, 15.90034952181462, 16.32896588729640, 17.08200938497298

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