Properties

Degree 1
Conductor 193
Sign $0.957 + 0.289i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.831 − 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (0.290 + 0.956i)5-s + (0.980 − 0.195i)6-s + (0.707 + 0.707i)7-s + (−0.195 − 0.980i)8-s + (0.707 + 0.707i)9-s + (0.773 + 0.634i)10-s + (0.471 + 0.881i)11-s + (0.707 − 0.707i)12-s + (−0.634 − 0.773i)13-s + (0.980 + 0.195i)14-s + (−0.0980 + 0.995i)15-s + (−0.707 − 0.707i)16-s + (−0.290 + 0.956i)17-s + ⋯
L(s,χ)  = 1  + (0.831 − 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (0.290 + 0.956i)5-s + (0.980 − 0.195i)6-s + (0.707 + 0.707i)7-s + (−0.195 − 0.980i)8-s + (0.707 + 0.707i)9-s + (0.773 + 0.634i)10-s + (0.471 + 0.881i)11-s + (0.707 − 0.707i)12-s + (−0.634 − 0.773i)13-s + (0.980 + 0.195i)14-s + (−0.0980 + 0.995i)15-s + (−0.707 − 0.707i)16-s + (−0.290 + 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.957 + 0.289i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.957 + 0.289i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $0.957 + 0.289i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (74, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ 0.957 + 0.289i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.624821331 + 0.6833829211i$
$L(\frac12,\chi)$  $\approx$  $4.624821331 + 0.6833829211i$
$L(\chi,1)$  $\approx$  2.530340648 + 0.04548675310i
$L(1,\chi)$  $\approx$  2.530340648 + 0.04548675310i

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

−26.72789657053818303405186349395, −25.36354566748149037036579109230, −24.83873743996278739236305343574, −23.95558466155036332241708769508, −23.4926492197762684151233532177, −21.68823533106505454189812072732, −21.19095014305556026682509624785, −20.2323369499772286746005984522, −19.42731477033308349403132372199, −17.8080949273965209361366887734, −16.91806637915581666477746447869, −16.04251077816980027597546098465, −14.72382370500887155492111752960, −13.974661017560107104582518832551, −13.35218584600769635976615119677, −12.31417133300715279831025811883, −11.21831944065570021535337742804, −9.28830456606144873188934818432, −8.50105506762320126203956206682, −7.46908081170483312586849331804, −6.46593781008591488875033447672, −4.88742871221634284537840380329, −4.12696913705398731244150957841, −2.656862574998169137851442818463, −1.231082691397195276283395359774, 1.95192798677930481210186664049, 2.5541050193155020087518105353, 3.868002469846385473719495722962, 4.93213782658408235167416652969, 6.292317557434742052452020612820, 7.56576857344543060144021746038, 8.99201353754217643440572069577, 10.17367801349105252111203957670, 10.81104760495811614283257964167, 12.20089085604759854278648159345, 13.149667569140323162026283971881, 14.443406032952777075052438160895, 14.89787162312767575182073282528, 15.38361646996892620646060892847, 17.32988082986652202883315471014, 18.57720039863981470777313877568, 19.36225741314604748965199628689, 20.32159284725177072844670713995, 21.25950111686406515420626786862, 21.953882350568737634010012694491, 22.688828239296283487787619037396, 24.01178939812233006021988194072, 25.13106794449555431289011216922, 25.54493573524360337973117514434, 27.06834920432533006056559527241

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