Properties

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

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 14-s + 16-s − 17-s − 19-s + 20-s + 22-s − 23-s + 25-s − 28-s − 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s + 41-s + 43-s + 44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(39\)    =    \(3 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{39} (38, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 39,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.776581434$
$L(\frac12,\chi)$  $\approx$  $2.776581434$
$L(\chi,1)$  $\approx$  2.012229726
$L(1,\chi)$  $\approx$  2.012229726

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−34.7455728898393599410470802854, −33.36575152698929719192568831969, −32.654851397097391171163639940629, −31.63378022587838460339783881324, −30.071211749866349843068044079398, −29.38995764131271612067805504454, −28.21670238714452437561428321609, −26.080358633423214338465013260801, −25.2354638079280302768014451571, −24.092599756848773483522062032142, −22.47568677187093845556963849738, −21.93315191768140742388488531843, −20.48894258051107947462916786866, −19.28509033192190916774412727849, −17.337565702284410625359178605047, −16.14464333650454194819340125922, −14.63812553931791531619782220267, −13.467960588076771502489339540526, −12.46016779683215014834642518471, −10.75293335203503761289374520253, −9.26505664657555325046029148131, −6.79945605063326105935862116884, −5.85202180427453258469290681861, −3.955373319394198948596812494220, −2.15367543022529411299025556090, 2.15367543022529411299025556090, 3.955373319394198948596812494220, 5.85202180427453258469290681861, 6.79945605063326105935862116884, 9.26505664657555325046029148131, 10.75293335203503761289374520253, 12.46016779683215014834642518471, 13.467960588076771502489339540526, 14.63812553931791531619782220267, 16.14464333650454194819340125922, 17.337565702284410625359178605047, 19.28509033192190916774412727849, 20.48894258051107947462916786866, 21.93315191768140742388488531843, 22.47568677187093845556963849738, 24.092599756848773483522062032142, 25.2354638079280302768014451571, 26.080358633423214338465013260801, 28.21670238714452437561428321609, 29.38995764131271612067805504454, 30.071211749866349843068044079398, 31.63378022587838460339783881324, 32.654851397097391171163639940629, 33.36575152698929719192568831969, 34.7455728898393599410470802854

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