Properties

Degree 1
Conductor 13
Sign $0.522 + 0.852i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s i·5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s − 12-s + 14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s i·18-s + ⋯
L(s,χ)  = 1  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s i·5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s − 12-s + 14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s i·18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.522 + 0.852i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.522 + 0.852i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.522 + 0.852i$
motivic weight  =  \(0\)
character  :  $\chi_{13} (2, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 13,\ (1:\ ),\ 0.522 + 0.852i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.346813704 + 0.7545968286i$
$L(\frac12,\chi)$  $\approx$  $1.346813704 + 0.7545968286i$
$L(\chi,1)$  $\approx$  1.300817583 + 0.5339369191i
$L(1,\chi)$  $\approx$  1.300817583 + 0.5339369191i

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

−42.916493459784617707701619768426, −41.575207017812764885467506475328, −40.884849005313018622653349190751, −39.446147257535379165436770301338, −37.98662174149648259326393740307, −36.59736365804954566584311360769, −34.36362583119418884864240426704, −33.73503213528638777331012671536, −31.3714238119598700960972215137, −30.44161532015156837717137813753, −29.31249399810574252421122238580, −27.82848702324585128669644429138, −25.307477164656216181300697786524, −23.7723130738260825346680194365, −22.707233933565961819535493352100, −21.193797014928129437987555228002, −19.12658986031807858311542402985, −17.95727240089126309718459523372, −15.19982307397033042107583736742, −13.68260616414035954960946604229, −11.976074282559220921024912566493, −10.74269657978003853822168308889, −7.21770260402982876163191488360, −5.37688302580705465622725578544, −2.34546853359506404994450241032, 4.24460934264584967159571245224, 5.57713196951261458498668689850, 8.28909241938397139268805863170, 10.93323919116728175068762537924, 12.71207036592429069533654996003, 14.662973932502194903070243897304, 16.25589362511344762130165698068, 17.25138541491035773839727275862, 20.65426041658851831825796728705, 21.36664226902833192750943599219, 23.25372235027411254080364224307, 24.26191061336298593641419872381, 26.22481109400446508001175625025, 27.78380944627266683281517490520, 29.444571537950115649748331238947, 31.38841268898555875278187245581, 32.53738083646613564275504133147, 33.61489336611419017475205563412, 34.84394463394565934266337377085, 36.81290632835271241146079108369, 38.90716595030295196315185881850, 39.85967535328406576141814214294, 40.70096083791469259116137271408, 42.76588365433364961049527879308, 43.838348329304751474850602875092

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