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Properties

Label 2-95-19.7-c1-0-2
Degree $2$
Conductor $95$
Sign $0.910 - 0.412i$
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 + 1.73i)3-s + (1 − 1.73i)4-s + (0.5 + 0.866i)5-s − 4·7-s + (−0.499 + 0.866i)9-s + 3·11-s + 3.99·12-s + (−1 + 1.73i)13-s + (−0.999 + 1.73i)15-s + (−1.99 − 3.46i)16-s + (−3 − 5.19i)17-s + (−3.5 + 2.59i)19-s + 1.99·20-s + (−4 − 6.92i)21-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.577 + 0.999i)3-s + (0.5 − 0.866i)4-s + (0.223 + 0.387i)5-s − 1.51·7-s + (−0.166 + 0.288i)9-s + 0.904·11-s + 1.15·12-s + (−0.277 + 0.480i)13-s + (−0.258 + 0.447i)15-s + (−0.499 − 0.866i)16-s + (−0.727 − 1.26i)17-s + (−0.802 + 0.596i)19-s + 0.447·20-s + (−0.872 − 1.51i)21-s + (−0.0999 + 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15103 + 0.248770i\)
\(L(\frac12)\) \(\approx\) \(1.15103 + 0.248770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (3.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.45727815431112247704379615372, −13.23941662520608353357695345489, −11.74685280654655406145570868760, −10.55286062112174331753025360208, −9.547593772168487256781364363024, −9.280394832524382960342480921885, −6.93957105375992591277788891737, −6.12009696796375254811739836851, −4.25245972936901947248198134348, −2.76150367300491270317309970663, 2.29937073169802967093938323200, 3.77337017171378786291803365910, 6.33318627269780152687854974749, 7.03575539981179466954492587892, 8.323530410914496101847811241859, 9.239927903917167007568240168427, 10.80327490361199919615475316806, 12.40959478732948953346559269490, 12.79406976587063842094663671018, 13.46187718623727369698285947222

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