Properties

Degree 1
Conductor $ 2^{3} \cdot 7 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 3-s + 5-s + 9-s − 11-s + 13-s + 15-s − 17-s + 19-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 37-s + 39-s − 41-s − 43-s + 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s + 61-s + 65-s + ⋯
L(s,χ)  = 1  + 3-s + 5-s + 9-s − 11-s + 13-s + 15-s − 17-s + 19-s + 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 37-s + 39-s − 41-s − 43-s + 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s + 61-s + 65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(56\)    =    \(2^{3} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{56} (13, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 56,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.447541817\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.447541817\)
\(L(\chi,1)\)  \(\approx\)  \(1.679251908\)
\(L(1,\chi)\)  \(\approx\)  \(1.679251908\)

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

−32.9010348270297627088892070871, −31.51944231943997184472652470117, −30.690000746945030200012261744824, −29.43000418345294518925026094548, −28.41414520960288211549091183390, −26.75231591592850034861396584812, −25.92960509169980967974778911023, −25.01828837110298869760597156425, −23.9105265517369983533216599615, −22.22697215961054858321994589585, −21.008822935727199961141204299343, −20.35867295442329041893126835339, −18.73788426990777456133268263405, −17.89852150528378451400371119022, −16.17944515506529237932446357855, −14.98534622000614502074559353283, −13.60126566450937089198195650280, −13.06130986325124306941443320343, −10.854841016737571399498258970083, −9.56977863791393047057245889649, −8.48938031650563658288632406702, −6.93325712626592541726906678836, −5.20981502796481458623915670533, −3.252991709700736450463186226578, −1.789478909894140664437254075447, 1.789478909894140664437254075447, 3.252991709700736450463186226578, 5.20981502796481458623915670533, 6.93325712626592541726906678836, 8.48938031650563658288632406702, 9.56977863791393047057245889649, 10.854841016737571399498258970083, 13.06130986325124306941443320343, 13.60126566450937089198195650280, 14.98534622000614502074559353283, 16.17944515506529237932446357855, 17.89852150528378451400371119022, 18.73788426990777456133268263405, 20.35867295442329041893126835339, 21.008822935727199961141204299343, 22.22697215961054858321994589585, 23.9105265517369983533216599615, 25.01828837110298869760597156425, 25.92960509169980967974778911023, 26.75231591592850034861396584812, 28.41414520960288211549091183390, 29.43000418345294518925026094548, 30.690000746945030200012261744824, 31.51944231943997184472652470117, 32.9010348270297627088892070871

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