↑

Properties

Label 1-193-193.35-r1-0-0
Degree $1$
Conductor $193$
Sign $0.0954 - 0.995i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.0954 - 0.995i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.0954 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314565013 - 1.194546254i\)
\(L(\frac12)\) \(\approx\) \(1.314565013 - 1.194546254i\)
\(L(1)\) \(\approx\) \(0.9692535958 - 0.4745975144i\)
\(L(1)\) \(\approx\) \(0.9692535958 - 0.4745975144i\)

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

Imaginary part of the first few zeros on the critical line

−27.21243864076637946778979414133, −26.04672576212199147049584572137, −25.343264603944775382984285920412, −24.416059034263924220571524080159, −23.61233367206698285531791684277, −22.2616495888265197775049565113, −21.06337062349041305810071162531, −20.05334967219582020125169770441, −19.24072854883205742179324131367, −18.601829022611459831538184451839, −17.406603834804709225025494443683, −16.08135758994286102474362568853, −15.39606500780103295490667387774, −14.73035932855583116642860159346, −13.77091564289605348061620296912, −11.90144448072216620778428730887, −10.98834577071131925165953798454, −9.85264430599909871556382261772, −8.4534518741786041564006836806, −8.34585042401316618911290599636, −7.07489053853335919789559582660, −5.63596195455268963262527265484, −4.161460707404497214716671837, −2.81099057322764261872827455926, −1.22463845246388466724828219806, 0.91070630728720219196507205208, 1.8662795331437860849672880841, 3.6443711325513979039864298294, 4.150277272815687335349264640229, 6.78107738416721585431196623789, 7.71716042467143774205197844070, 8.38312166255779324877060314384, 9.37985427167445278571459219200, 10.60696145471142935746310071844, 11.83033811493213440047229243246, 12.44709079633497467834478494542, 13.86546428693927586826638157183, 14.79017823488215015408166671641, 16.12852973720617934590121914986, 16.96100476601237551668041346251, 18.27814902969203065657664309158, 18.96792221594007161873106299396, 19.95803427419722820585906823741, 20.49102759247969261440381564317, 21.25132632080396013156604175385, 22.89680093632731441486423323689, 23.94548690580080703356109470472, 24.86576828325513299287355506049, 25.835428143043872418417408745151, 26.70697765280899172424048580613

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