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Properties

Label 1-95-95.48-r0-0-0
Degree $1$
Conductor $95$
Sign $0.941 + 0.335i$
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  + (−0.342 − 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.984 − 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.984 + 0.173i)22-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.984 − 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.984 + 0.173i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.941 + 0.335i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.941 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4490353055 + 0.07760189377i\)
\(L(\frac12)\) \(\approx\) \(0.4490353055 + 0.07760189377i\)
\(L(1)\) \(\approx\) \(0.5549879601 - 0.09411049785i\)
\(L(1)\) \(\approx\) \(0.5549879601 - 0.09411049785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−29.82219680620842477337674611952, −28.87672929854960726083298530385, −28.0121639258191842598096623265, −26.8669527926931138781205263981, −26.1671306313105977813337118700, −24.864184898096065759496786546692, −23.72561886940633179855069100216, −23.02373287865917586096379077806, −22.17276871763267165691440586353, −20.709678685190640726205827897137, −18.94184199763744843197177445387, −18.350735606510127385353928420509, −16.93252688328279929821862818963, −16.321415417322182016536011507348, −15.51393744937989793069202518477, −13.83282042459143820102096160952, −12.89958764212157966153112472615, −11.13557651136480454330019045331, −10.15964931951932118148280261197, −8.94570124425910943279407910241, −7.33018368667236355020090854216, −6.30488585699673124369554088235, −5.32299476333041799659366314865, −3.803224813090004586609660730, −0.64919070436387722148670700970, 1.576103939022928853235512636069, 3.36569320300418268915494610642, 4.9407235616927630845185376981, 6.346987681084515790920164873157, 7.92516977097472179202157566085, 9.50217022969609462244497371687, 10.44508412081653432228800985660, 11.54505086900367973766454007588, 12.66893064994859164990837659627, 13.2371272051879535393779088991, 15.37873690842763782884932975538, 16.57810975847017222419335684017, 17.66815017675738839475405129096, 18.55016769134946511222137326061, 19.435617669717698481762859463193, 20.85562205269317218301106739759, 21.79933916189447404461387809636, 22.83437601802310318923041335034, 23.462039168191209580577316333626, 25.287258701274419376677126775837, 26.209480155967613111100965203046, 27.65122148396736540323838214640, 28.35894181879409791756091635743, 28.965278594059278275830143525211, 30.07890944987896539769164518634

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