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Properties

Label 2-95-19.9-c1-0-3
Degree $2$
Conductor $95$
Sign $0.998 + 0.0504i$
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  + (0.0366 + 0.207i)2-s + (0.0612 − 0.0513i)3-s + (1.83 − 0.668i)4-s + (−0.939 − 0.342i)5-s + (0.0129 + 0.0108i)6-s + (0.843 − 1.46i)7-s + (0.417 + 0.722i)8-s + (−0.519 + 2.94i)9-s + (0.0366 − 0.207i)10-s + (1.44 + 2.50i)11-s + (0.0781 − 0.135i)12-s + (−4.95 − 4.15i)13-s + (0.334 + 0.121i)14-s + (−0.0750 + 0.0273i)15-s + (2.86 − 2.40i)16-s + (0.518 + 2.94i)17-s + ⋯
L(s)  = 1  + (0.0259 + 0.146i)2-s + (0.0353 − 0.0296i)3-s + (0.918 − 0.334i)4-s + (−0.420 − 0.152i)5-s + (0.00527 + 0.00442i)6-s + (0.318 − 0.552i)7-s + (0.147 + 0.255i)8-s + (−0.173 + 0.982i)9-s + (0.0115 − 0.0657i)10-s + (0.435 + 0.753i)11-s + (0.0225 − 0.0390i)12-s + (−1.37 − 1.15i)13-s + (0.0894 + 0.0325i)14-s + (−0.0193 + 0.00705i)15-s + (0.715 − 0.600i)16-s + (0.125 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.998 + 0.0504i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.998 + 0.0504i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12228 - 0.0283255i\)
\(L(\frac12)\) \(\approx\) \(1.12228 - 0.0283255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (4.34 + 0.288i)T \)
good2 \( 1 + (-0.0366 - 0.207i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.0612 + 0.0513i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.843 + 1.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.44 - 2.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.95 + 4.15i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.518 - 2.94i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.75 - 2.82i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.26 + 7.14i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.02 - 3.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
41 \( 1 + (-4.17 + 3.50i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.01 - 1.82i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.286 + 1.62i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.79 - 0.653i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.616 - 3.49i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.42 + 2.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.393 + 2.23i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (9.79 + 3.56i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.08 + 0.907i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.84 - 1.54i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.70 + 9.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.74 + 6.50i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.71 + 9.71i)T + (-91.1 + 33.1i)T^{2} \)
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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

−14.26974334710863659685868511998, −12.80176215929988167574289484331, −11.85680717551057067114966327979, −10.70424032180262221143739847484, −9.984782345602093186747353919523, −7.952205058554771187561167474300, −7.45807556881787800628190613503, −5.87492448938355178049860210400, −4.43935927979513493699991009713, −2.23510460173794844595056560297, 2.49247590167177974828871163261, 4.09361556521959449088007602872, 6.11479637166034230155448422455, 7.12468115744090198594230657842, 8.443599620668874221262387002899, 9.643316545452058614816440773751, 11.15268355017876212631884963138, 11.89519057778589336333267546369, 12.48128278493120836234097982236, 14.37789409436087866117981202887

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