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Properties

Label 1-193-193.74-r1-0-0
Degree $1$
Conductor $193$
Sign $0.957 + 0.289i$
Analytic cond. $20.7407$
Root an. cond. $20.7407$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.624821331 + 0.6833829211i\)
\(L(\frac12)\) \(\approx\) \(4.624821331 + 0.6833829211i\)
\(L(1)\) \(\approx\) \(2.530340648 + 0.04548675310i\)
\(L(1)\) \(\approx\) \(2.530340648 + 0.04548675310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + (0.831 - 0.555i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (0.290 + 0.956i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.471 + 0.881i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (-0.290 + 0.956i)T \)
19 \( 1 + (-0.956 + 0.290i)T \)
23 \( 1 + (0.831 - 0.555i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (-0.555 - 0.831i)T \)
37 \( 1 + (0.995 + 0.0980i)T \)
41 \( 1 + (-0.471 - 0.881i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.773 - 0.634i)T \)
53 \( 1 + (-0.634 - 0.773i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (-0.956 - 0.290i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (0.956 - 0.290i)T \)
73 \( 1 + (-0.995 + 0.0980i)T \)
79 \( 1 + (0.881 - 0.471i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (0.634 - 0.773i)T \)
97 \( 1 + (0.831 + 0.555i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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