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Properties

Label 1-95-95.32-r0-0-0
Degree $1$
Conductor $95$
Sign $0.620 + 0.784i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5183360401 + 0.2510204472i\)
\(L(\frac12)\) \(\approx\) \(0.5183360401 + 0.2510204472i\)
\(L(1)\) \(\approx\) \(0.6163514984 + 0.1470263722i\)
\(L(1)\) \(\approx\) \(0.6163514984 + 0.1470263722i\)

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

−29.91187023885991900149796262770, −28.81876369636098811473473110596, −27.984902720517875173422488298366, −27.19283014982389376740263753637, −25.80215991846866759036485776725, −24.745225766170684651754377438, −24.11936115266296606453845920817, −22.99704003218600453438138085171, −21.42412913322953873790268588662, −20.365965567869492819191658553, −18.79634953553352402164864673101, −18.46066671549034201224514739888, −17.3600791955824843373128710181, −16.391612049094942565872522919207, −15.18635374294404501308990466955, −13.70440376641731970947572799235, −12.1513471889552563856377097966, −11.26216853656172721800962956035, −10.26223714563312612648697045401, −8.37383841800383148554400697193, −7.865367134676796216611225378499, −6.25154879184058784231099503286, −5.37498408030420870674791793606, −2.58559508807980371139347458271, −1.02389473866116643908905561091, 1.52739099146220105774512610719, 3.65525240643549863543663041837, 5.17290012682726943699880477845, 6.76978561925717857871655246212, 8.05935156731629719519102601422, 9.44147471882633239516471809625, 10.44039368640851735428700942571, 11.30875853469431282343841771342, 12.35975370533413939832716496945, 14.39568685305524205691094606357, 15.622430550527344244065866692186, 16.55526355409757455149933694786, 17.56625770918958327533130048214, 18.274414168142221258827601277258, 19.83185464706985224817092339940, 20.96600161050649713269604467915, 21.39922518736527903546393824648, 23.151788943276407786718582711088, 23.96567503674043622854767226229, 25.578256137801433176541075440388, 26.37296688176562971594386356711, 27.451132744307893329918867159570, 28.04129762474790278822868215124, 28.996069332149824535536274171609, 30.081164538136361974381127622318

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