Properties

Label 1-95-95.4-r0-0-0
Degree $1$
Conductor $95$
Sign $0.845 - 0.533i$
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  + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 18-s + (0.766 + 0.642i)21-s + (0.939 − 0.342i)22-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.939 − 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 + 0.342i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 18-s + (0.766 + 0.642i)21-s + (0.939 − 0.342i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9398706067 - 0.2715728501i\)
\(L(\frac12)\) \(\approx\) \(0.9398706067 - 0.2715728501i\)
\(L(1)\) \(\approx\) \(0.9711185436 - 0.2403333339i\)
\(L(1)\) \(\approx\) \(0.9711185436 - 0.2403333339i\)

Euler product

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

−30.32430211765681752144216858519, −29.14721766861342294081216617781, −27.768708591573862931623233062181, −26.96702813027569224056367961806, −26.20536764841045111996248603327, −25.34709634622700110931144559498, −24.17124012986115197345622417976, −23.42619605420842368233325533072, −21.617479066342630354877493899905, −20.46165552794910168816674527135, −19.674898128549066750350143525282, −18.55885374896452736427118631047, −17.44183526655684360950897335081, −16.152913253466393382579381845379, −15.39397251618683506523397737986, −14.10862409263437016671922605404, −13.42367971616142884971824908002, −10.9657039535533141687097000865, −10.29752261999715978574497880716, −8.760558005253667685418250992376, −8.128462746416670406499429447750, −6.87967715361692860347076227819, −5.19274537513293561938117087424, −3.57624278820693916662090787617, −1.59318682945951403135706064714, 1.781918095424741201044919671263, 2.75499306371509766508661165391, 4.39604106800745702544454008547, 6.72914088820985498738093310352, 8.061326839581362901472910714738, 8.83376449997962239896239939936, 9.92409096911201095284465679147, 11.40836842034528263468777100858, 12.51566265918008443172662491001, 13.55669120763676759857589463150, 15.05262767817595542318966495802, 16.0662243727242049694642614291, 17.9006710707862972932836377997, 18.34818824434334560081461465648, 19.43153064219098321138673010961, 20.6337345064827198194405066263, 21.06682031404766613594235927905, 22.49946780374511918975377623114, 24.19412134872849662170997278561, 25.21365622120182109876122672198, 25.96349172518357766324328934323, 26.95859523148855408773325357765, 28.11325515933195740608134640793, 28.924934422468191092516807476046, 30.28903844525284651030229136206

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