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Properties

Label 1-95-95.12-r0-0-0
Degree $1$
Conductor $95$
Sign $0.557 + 0.830i$
Analytic cond. $0.441178$
Root an. cond. $0.441178$
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  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + i·7-s + i·8-s + (0.5 + 0.866i)9-s + 11-s − i·12-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + i·18-s + (0.5 − 0.866i)21-s + (0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + i·7-s + i·8-s + (0.5 + 0.866i)9-s + 11-s − i·12-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + i·18-s + (0.5 − 0.866i)21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.557 + 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129294239 + 0.6018306487i\)
\(L(\frac12)\) \(\approx\) \(1.129294239 + 0.6018306487i\)
\(L(1)\) \(\approx\) \(1.237153487 + 0.4078457326i\)
\(L(1)\) \(\approx\) \(1.237153487 + 0.4078457326i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−29.99492928893094562597283834495, −29.12422358282830922704806344371, −28.15517008361133444718887736131, −27.24936453818301426789820624228, −25.94316240522224179529648418433, −24.22975704917606257283599900630, −23.59096954562611421012123576600, −22.530372503353909304726587518880, −21.84316811191481990602730995796, −20.64515156151628572172011153229, −19.81813409470264650937503387810, −18.31861497202367734120860378939, −16.93018229541846222188386919873, −16.039662564545102978857570133491, −14.74979702378020186153167768088, −13.648292356967382192137407212580, −12.41617669551249468511046116282, −11.2413923262526246781017467718, −10.59212734976097782257132193194, −9.2712328355445280820611775468, −6.88267575149571858821591056468, −6.01286448643839728083896325752, −4.411475228490215365729723999572, −3.76905919869107562562520930209, −1.38076989183911729290729782057, 2.12168953533703915060002234214, 4.01486530078649096320153656735, 5.53883189997331943956791132754, 6.22627842989616113176444260048, 7.50189504382612886789248678652, 8.94426772925586634557936596092, 11.07861748194720656442975003939, 11.89778727636431091604236593539, 12.85672552706959769173290699435, 13.95479978011947570906787566027, 15.38220228120943019720859848479, 16.213648823526213246303219253114, 17.45361192748791096998560015792, 18.312897979953532995961969590612, 19.82103031490559386182220620738, 21.31540666249970828164133537695, 22.29503139437823063025070850970, 22.8546695993409903405550483752, 24.172203516759550902331575707107, 24.80586555465007854777817453999, 25.74848813801418354812251803191, 27.418420661209064319135099977290, 28.4041401195211641079807229399, 29.56357934416727538369667759709, 30.41831252101117578586030439350

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