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Properties

Label 1-95-95.78-r0-0-0
Degree $1$
Conductor $95$
Sign $0.121 - 0.992i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.121 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103125075 - 0.9767328988i\)
\(L(\frac12)\) \(\approx\) \(1.103125075 - 0.9767328988i\)
\(L(1)\) \(\approx\) \(1.258591850 - 0.7524682387i\)
\(L(1)\) \(\approx\) \(1.258591850 - 0.7524682387i\)

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

−30.79500207964569393922572964021, −29.862805429561884969134757583240, −27.96576147836616062835333074633, −26.789988726611989147198589354308, −26.38232584799329504322725479338, −25.04094215343365164196843593436, −24.43938351906292808294082211232, −23.31910589763717760459694284818, −22.06264562322739181883648448429, −20.947459591223071415853444194058, −20.076017174583696989351029580320, −18.49479159088517958922764298868, −17.525711911618405010690681482236, −16.23633066386649088825523067239, −15.09576969281015327692007027822, −14.37206471225180658097090114400, −13.44087394625102217409865532594, −12.175968694859511199726525069282, −10.19898966033617498577877033579, −9.01245603907363898247628796918, −7.755015152985089452321376293249, −7.116530873954817859218798006997, −5.01531309320698912759062665302, −4.18166848426205645478577207523, −2.41674009102894215488032664178, 1.75612098497073911975660428288, 2.87321989011977894196903684514, 4.2754464697598438098067750637, 5.69431428592210109942815756582, 7.83938189414794400010126502288, 8.810150410922888160638552910045, 10.029630779263157358532516685995, 11.31749928772834583226354477048, 12.533386920220810436529981906298, 13.569643429108263841333316687099, 14.57589871673087094497707645998, 15.40566862949329428997726804464, 17.50627448192037897670425508087, 18.64348511640401757953357100779, 19.41395477001431548983913796644, 20.49596193682872351244643635540, 21.38041356131302482292934663800, 22.142730635438750303134450661869, 24.05711624783341647873300263020, 24.26943313918781337051107803862, 25.961924209181674640221028550355, 27.01533136195015498872892954876, 27.84758240834324580066164501189, 29.22726506959926268023749086556, 30.059814026784113000268959446359

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