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Properties

Label 1-195-195.137-r1-0-0
Degree $1$
Conductor $195$
Sign $-0.547 + 0.836i$
Analytic cond. $20.9556$
Root an. cond. $20.9556$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + 8-s + (0.866 − 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)28-s + (−0.5 − 0.866i)29-s + i·31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s + 8-s + (0.866 − 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)28-s + (−0.5 − 0.866i)29-s + i·31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.547 + 0.836i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ -0.547 + 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1123181459 + 0.2077069962i\)
\(L(\frac12)\) \(\approx\) \(0.1123181459 + 0.2077069962i\)
\(L(1)\) \(\approx\) \(0.6228646995 - 0.1301783901i\)
\(L(1)\) \(\approx\) \(0.6228646995 - 0.1301783901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−26.26672272431853036574783959130, −25.73172027107456635063568326716, −24.48478048138764301414609381193, −23.89074455291897864928364732041, −22.676654062224265819887411910413, −22.16261398638000466568804838092, −20.21443171155630875139939753369, −19.80900853093391153816020757158, −18.58672940519466777595369245481, −17.55002861182770255738652257658, −16.82620200227263081066302766450, −15.90133972302188493300562263608, −14.86209016347607835198198022435, −13.89597107140800255593546077064, −12.97809972822616813662526274955, −11.40128560169173170317587957690, −10.17985750592364644644087209668, −9.41047093542982165285275367854, −8.23142816034040198490688991690, −7.02118018584722090990207360346, −6.40463866155554328108653553135, −4.88469309300590812204669969660, −3.7625645222961850347546307953, −1.610308128725387502722507489668, −0.09842489248525825853276923031, 1.5882980907561753626477428118, 2.90042527539398158265065568766, 3.97145976097923615227996104882, 5.53659388240050288009533245364, 6.93652889408861790943984559104, 8.361016578312648940880405697724, 9.20593846769272717570679456608, 10.05755840626994324844829656406, 11.4904079275151252121486786, 11.9903666460508408631367440185, 13.19481869240432063850357603822, 14.15649417592587666848241883845, 15.6838665886173075345441405422, 16.5614999088861444897912831778, 17.74802755204531287365065515756, 18.54412485846791694209146224910, 19.4793715205471286456633970809, 20.20648742014305927475170324035, 21.44862740808754729115027832863, 22.133830923592228036825732460602, 22.88594784032161284459468613648, 24.53345781206986946692089971378, 25.26773610137560311958846650492, 26.405241187089289038495537771328, 27.16804414928909646502232072805

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