Properties

Label 1-95-95.22-r0-0-0
Degree $1$
Conductor $95$
Sign $0.979 + 0.203i$
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  + (0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + i·18-s + (−0.173 − 0.984i)21-s + (−0.642 − 0.766i)22-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + i·18-s + (−0.173 − 0.984i)21-s + (−0.642 − 0.766i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.979 + 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872581779 + 0.1927671973i\)
\(L(\frac12)\) \(\approx\) \(1.872581779 + 0.1927671973i\)
\(L(1)\) \(\approx\) \(1.838692773 + 0.1274764926i\)
\(L(1)\) \(\approx\) \(1.838692773 + 0.1274764926i\)

Euler product

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

−30.46198814074760048596213738356, −29.2822798264595264976187778301, −28.65165098613031230086737917449, −26.632453224058986899377035753760, −25.51403657497808358996520939323, −25.00675715110188857065329376024, −23.90651225877103055645157690669, −22.856079740098656328042335457124, −21.97360888663577720809084172648, −20.46465511416569667537890581697, −19.88427169479194218975888137401, −18.58472984312856902353406358816, −17.28551998311209163670323973528, −15.66199285377159154856810057032, −15.00144156952645036504489420355, −13.711591672133202478429149178314, −12.710030576532295790227537754759, −12.18880689077876370797001311560, −10.33157581797586164342841406784, −8.70776911578193548721151721662, −7.30656765843295031237701236702, −6.45675950552679809155241353225, −4.93849685011049851135768831656, −3.17684486524748081578099920894, −2.253504331931098602562747865621, 2.442516123299454621555440937951, 3.59733237669637209188322570108, 4.668377472672580618799167832060, 6.164239739083310320957733936968, 7.60210414666074883314035644252, 9.32348745106667839689336299318, 10.42352110687838852829670602370, 11.513002548531072821642746925, 13.21883543704660395829562328940, 13.762826536745675255444201411904, 15.057810853994312455302728401749, 15.967682138243349710038256569416, 16.84281317016328924991155021665, 19.19235863796188257490433810291, 19.7263724250016436163428366167, 20.96027158433039401103969251877, 21.75080390713193268289611742871, 22.64997420602986967580393228094, 23.867302513918903462714815616214, 24.92929196444987797893796090577, 26.12392311403299272096626068332, 26.819406438470051887779239680997, 28.467702320017509352337355439883, 29.28186867090332138439439688454, 30.41029597575508296776373213810

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