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Properties

Label 1-195-195.77-r0-0-0
Degree $1$
Conductor $195$
Sign $0.850 + 0.525i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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  + i·2-s − 4-s − i·7-s − i·8-s + 11-s + 14-s + 16-s − i·17-s + 19-s + i·22-s + i·23-s + i·28-s + 29-s − 31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·7-s − i·8-s + 11-s + 14-s + 16-s − i·17-s + 19-s + i·22-s + i·23-s + i·28-s + 29-s − 31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.042651484 + 0.2961954367i\)
\(L(\frac12)\) \(\approx\) \(1.042651484 + 0.2961954367i\)
\(L(1)\) \(\approx\) \(0.9681547780 + 0.3056792674i\)
\(L(1)\) \(\approx\) \(0.9681547780 + 0.3056792674i\)

Euler product

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

−27.18638504888540813852023325671, −26.164870571327705840586408844064, −25.01294366412670048282379439288, −24.0338842275804069773725705779, −22.73980498366050623236020607409, −22.04322518135186104239680187141, −21.30549192342264427493631354735, −20.1545607808731480771971022620, −19.35839112513440103569544118360, −18.43833733568208351787976390731, −17.59269586586615126408087077559, −16.40918768298628798181869868594, −14.9609867141209931161561030862, −14.18242425046637224235566065590, −12.87190874854631587222830101230, −12.08252944279473018240870753360, −11.25054668158192371999998864570, −10.00561112932298363521414623864, −9.05025448445883359954022534526, −8.194427643312040777171495861788, −6.37906802666431732599902329952, −5.172167168955345056969809535038, −3.90758286640424993906223379324, −2.70049951807787762911257354723, −1.419608850470860351740105327698, 1.05108476893297549020533879296, 3.457513670148670781460928298125, 4.49446679428091348585861793046, 5.721339227691626991040418845739, 7.001176698368060475021281469951, 7.59338740418120221988209685825, 9.05362034526615340812585814198, 9.82543093497110025234230408155, 11.24196164207500823788736762932, 12.55805064332003145098053423489, 13.916747695805673454990076052954, 14.16072051341515132662623091870, 15.60949772832728232571651601064, 16.425886439445214269546697021282, 17.32840387708839959394329202411, 18.10787066424541482637780474549, 19.386560189533656008767179878768, 20.27411376851488466431254256077, 21.660519236684261552539685112814, 22.646245821092059380747744869065, 23.33221596892182617090616608374, 24.35199878477369420204789167863, 25.1303068810831571661279834652, 26.08531520448510263145217730997, 27.08257215027781032186017854835

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