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Properties

Label 2-95-95.18-c1-0-5
Degree $2$
Conductor $95$
Sign $0.935 - 0.352i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 1.10i)2-s + (1.29 − 1.29i)3-s + 0.460i·4-s + (−2.17 + 0.539i)5-s + 2.87·6-s + (−1.53 + 1.53i)7-s + (1.70 − 1.70i)8-s − 0.369i·9-s + (−3.00 − 1.80i)10-s − 2.63·11-s + (0.598 + 0.598i)12-s + (−1.29 + 1.29i)13-s − 3.41·14-s + (−2.11 + 3.51i)15-s + 4.70·16-s + (−0.709 + 0.709i)17-s + ⋯
L(s)  = 1  + (0.784 + 0.784i)2-s + (0.749 − 0.749i)3-s + 0.230i·4-s + (−0.970 + 0.241i)5-s + 1.17·6-s + (−0.581 + 0.581i)7-s + (0.603 − 0.603i)8-s − 0.123i·9-s + (−0.950 − 0.572i)10-s − 0.793·11-s + (0.172 + 0.172i)12-s + (−0.359 + 0.359i)13-s − 0.912·14-s + (−0.546 + 0.907i)15-s + 1.17·16-s + (−0.172 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45177 + 0.264687i\)
\(L(\frac12)\) \(\approx\) \(1.45177 + 0.264687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.17 - 0.539i)T \)
19 \( 1 + (-3.41 + 2.70i)T \)
good2 \( 1 + (-1.10 - 1.10i)T + 2iT^{2} \)
3 \( 1 + (-1.29 + 1.29i)T - 3iT^{2} \)
7 \( 1 + (1.53 - 1.53i)T - 7iT^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + (1.29 - 1.29i)T - 13iT^{2} \)
17 \( 1 + (0.709 - 0.709i)T - 17iT^{2} \)
23 \( 1 + (1.53 + 1.53i)T + 23iT^{2} \)
29 \( 1 + 1.84T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + (-3.51 - 3.51i)T + 37iT^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 + (-8.21 - 8.21i)T + 43iT^{2} \)
47 \( 1 + (6.80 - 6.80i)T - 47iT^{2} \)
53 \( 1 + (6.11 - 6.11i)T - 53iT^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (1.67 + 1.67i)T + 67iT^{2} \)
71 \( 1 - 3.99iT - 71T^{2} \)
73 \( 1 + (7.38 + 7.38i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (-2.51 - 2.51i)T + 83iT^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (9.14 + 9.14i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.10647001042372092069437028996, −13.18051832332947557637164747256, −12.46199723827172079144962176764, −11.04892266547777370812945699615, −9.482199019106267081744660323861, −7.948688296758890785869121834333, −7.33340096001609164554316514813, −6.08841879942456816381292873769, −4.54030521939753742955195686017, −2.82303812005383444580189304951, 3.08745966745244445129564652589, 3.82573071683239209805280598713, 5.03410589910246841978363843212, 7.39459982067165478552272712088, 8.423700022845485137119147222443, 9.854387799848741872853235186133, 10.79148032010042099222737031935, 12.00844315936651798671069967575, 12.81037961861867246001741142464, 13.84643036629968799097926501685

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