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Properties

Label 2-95-95.4-c1-0-4
Degree $2$
Conductor $95$
Sign $0.799 + 0.600i$
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  + (0.358 − 0.984i)2-s + (−0.523 + 0.0922i)3-s + (0.691 + 0.580i)4-s + (2.23 + 0.0926i)5-s + (−0.0966 + 0.548i)6-s + (−2.37 − 1.37i)7-s + (2.63 − 1.52i)8-s + (−2.55 + 0.929i)9-s + (0.891 − 2.16i)10-s + (−0.416 − 0.721i)11-s + (−0.415 − 0.239i)12-s + (0.601 + 0.106i)13-s + (−2.19 + 1.84i)14-s + (−1.17 + 0.157i)15-s + (−0.239 − 1.35i)16-s + (−1.65 + 4.54i)17-s + ⋯
L(s)  = 1  + (0.253 − 0.695i)2-s + (−0.302 + 0.0532i)3-s + (0.345 + 0.290i)4-s + (0.999 + 0.0414i)5-s + (−0.0394 + 0.223i)6-s + (−0.896 − 0.517i)7-s + (0.930 − 0.537i)8-s + (−0.851 + 0.309i)9-s + (0.281 − 0.684i)10-s + (−0.125 − 0.217i)11-s + (−0.119 − 0.0692i)12-s + (0.166 + 0.0294i)13-s + (−0.587 + 0.493i)14-s + (−0.304 + 0.0407i)15-s + (−0.0598 − 0.339i)16-s + (−0.401 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.799 + 0.600i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.799 + 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11078 - 0.370714i\)
\(L(\frac12)\) \(\approx\) \(1.11078 - 0.370714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.0926i)T \)
19 \( 1 + (4.35 + 0.175i)T \)
good2 \( 1 + (-0.358 + 0.984i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.523 - 0.0922i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.37 + 1.37i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.416 + 0.721i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.601 - 0.106i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.65 - 4.54i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.41 - 2.87i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.73 - 1.35i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.46 + 5.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.33iT - 37T^{2} \)
41 \( 1 + (-0.923 - 5.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.72 + 8.01i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.16 - 3.19i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.78 + 10.4i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-9.41 - 3.42i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.94 - 5.83i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.73 - 10.2i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.519 - 0.435i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (6.90 - 1.21i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.604 + 3.42i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.30 - 2.48i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.02 + 5.79i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.25 + 11.6i)T + (-74.3 - 62.3i)T^{2} \)
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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

−13.28828007219007026527794338031, −13.13363367604596883889822704696, −11.68515469735801165779234977588, −10.69311468000793510380211602178, −9.991982284195746435178616106519, −8.456751614302426042495454255788, −6.79625023744108769264490521577, −5.77937624631074079566457825309, −3.83829746287253239898675041780, −2.30160071333362868664815173823, 2.53120768778090713016460478818, 5.10891835690588552618061389957, 6.13827242286301481705831030173, 6.74944555780745769260838060983, 8.592001462815371084832356951861, 9.762145209819457298909434159449, 10.83915955573337868589985409674, 12.08808664417561710670984033256, 13.28115825300814417682977997992, 14.19007542566389157610018468116

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