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Properties

Label 2-95-19.11-c1-0-1
Degree $2$
Conductor $95$
Sign $-0.813 - 0.582i$
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  + (−0.610 + 1.05i)2-s + (−1.14 + 1.97i)3-s + (0.253 + 0.439i)4-s + (0.5 − 0.866i)5-s + (−1.39 − 2.41i)6-s − 1.28·7-s − 3.06·8-s + (−1.11 − 1.92i)9-s + (0.610 + 1.05i)10-s + 0.285·11-s − 1.15·12-s + (2.5 + 4.33i)13-s + (0.785 − 1.35i)14-s + (1.14 + 1.97i)15-s + (1.36 − 2.36i)16-s + (3.11 − 5.40i)17-s + ⋯
L(s)  = 1  + (−0.431 + 0.748i)2-s + (−0.659 + 1.14i)3-s + (0.126 + 0.219i)4-s + (0.223 − 0.387i)5-s + (−0.569 − 0.987i)6-s − 0.485·7-s − 1.08·8-s + (−0.370 − 0.641i)9-s + (0.193 + 0.334i)10-s + 0.0859·11-s − 0.334·12-s + (0.693 + 1.20i)13-s + (0.209 − 0.363i)14-s + (0.295 + 0.510i)15-s + (0.341 − 0.590i)16-s + (0.756 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.813 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208091 + 0.647957i\)
\(L(\frac12)\) \(\approx\) \(0.208091 + 0.647957i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 - 3.22i)T \)
good2 \( 1 + (0.610 - 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.14 - 1.97i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 0.285T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.11 + 5.40i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.61 - 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.642 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + (-0.420 + 0.728i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.47 + 4.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.86 + 4.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.18 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.86 - 4.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.492 - 0.853i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.46 + 2.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.382 + 0.661i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.72 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.66T + 83T^{2} \)
89 \( 1 + (-8.01 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.87 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.78104861279970454496830238553, −13.48802447675432840288618879059, −11.99877560153424103008135092915, −11.27590737200906454864819151397, −9.663590435919261005952754980624, −9.259588169779716538358223148758, −7.68076629521017739174448342339, −6.33393031207966910796482894119, −5.21132130397785845921866985352, −3.60568901642395268058536249356, 1.12371398621672147271265983892, 2.98324412847534021062814642174, 5.82228492076204796117938771273, 6.46573016481779933894531081485, 7.903599891186546234136024573448, 9.469812864236906960878360996511, 10.61806225798565234454106442283, 11.30595408627351100890932710096, 12.56953111495011084103795483964, 13.00755121947151701649398360770

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