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Properties

Label 2-95-95.49-c1-0-2
Degree $2$
Conductor $95$
Sign $0.967 - 0.253i$
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  + (0.408 + 0.235i)2-s + (0.900 + 0.520i)3-s + (−0.888 − 1.53i)4-s + (1.47 + 1.67i)5-s + (0.245 + 0.425i)6-s + 1.17i·7-s − 1.78i·8-s + (−0.958 − 1.66i)9-s + (0.208 + 1.03i)10-s + 0.713·11-s − 1.84i·12-s + (−3.55 + 2.05i)13-s + (−0.277 + 0.480i)14-s + (0.459 + 2.28i)15-s + (−1.35 + 2.34i)16-s + (−2.21 − 1.27i)17-s + ⋯
L(s)  = 1  + (0.288 + 0.166i)2-s + (0.520 + 0.300i)3-s + (−0.444 − 0.769i)4-s + (0.661 + 0.750i)5-s + (0.100 + 0.173i)6-s + 0.444i·7-s − 0.630i·8-s + (−0.319 − 0.553i)9-s + (0.0659 + 0.327i)10-s + 0.215·11-s − 0.533i·12-s + (−0.985 + 0.569i)13-s + (−0.0741 + 0.128i)14-s + (0.118 + 0.588i)15-s + (−0.339 + 0.587i)16-s + (−0.536 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.967 - 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23074 + 0.158782i\)
\(L(\frac12)\) \(\approx\) \(1.23074 + 0.158782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.47 - 1.67i)T \)
19 \( 1 + (1.57 + 4.06i)T \)
good2 \( 1 + (-0.408 - 0.235i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.900 - 0.520i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.17iT - 7T^{2} \)
11 \( 1 - 0.713T + 11T^{2} \)
13 \( 1 + (3.55 - 2.05i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.21 + 1.27i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.525 + 0.303i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.429 + 0.744i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 9.38iT - 37T^{2} \)
41 \( 1 + (-2.06 + 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.76 - 5.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.10 + 5.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.31 - 2.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.12 - 5.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.27 - 3.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.48 - 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.58 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.7 + 6.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 - 5.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-7.98 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.4 + 8.35i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.16432175717260961950190841974, −13.43204421713602257448287197782, −11.94619459305611533682540265172, −10.61798250281499905525037793496, −9.509303155160560054883114252388, −8.982078602410071207326878825497, −6.93376187291441956828161865353, −5.92463776722315823790812204697, −4.47884796933604235881539450925, −2.61610862242062314380527624568, 2.42156377301203964317270217254, 4.21555640223505984898769316635, 5.52390034395521762002462980130, 7.47867622812743580660950410577, 8.391308280150624689601056897885, 9.379532920517666341483083473836, 10.75433752918938850431395352398, 12.33948376270172223086483194828, 12.89169348738297921232402202012, 13.85286396205021266176377472026

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