Properties

Label 2-95-19.4-c1-0-2
Degree $2$
Conductor $95$
Sign $0.521 - 0.853i$
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  + (1.12 + 0.408i)2-s + (0.394 + 2.23i)3-s + (−0.439 − 0.369i)4-s + (−0.766 + 0.642i)5-s + (−0.471 + 2.67i)6-s + (1.09 − 1.90i)7-s + (−1.53 − 2.66i)8-s + (−2.03 + 0.741i)9-s + (−1.12 + 0.408i)10-s + (1.41 + 2.44i)11-s + (0.652 − 1.13i)12-s + (0.708 − 4.01i)13-s + (2.00 − 1.68i)14-s + (−1.74 − 1.46i)15-s + (−0.437 − 2.48i)16-s + (−0.359 − 0.130i)17-s + ⋯
L(s)  = 1  + (0.793 + 0.288i)2-s + (0.227 + 1.29i)3-s + (−0.219 − 0.184i)4-s + (−0.342 + 0.287i)5-s + (−0.192 + 1.09i)6-s + (0.415 − 0.718i)7-s + (−0.543 − 0.941i)8-s + (−0.678 + 0.247i)9-s + (−0.354 + 0.129i)10-s + (0.426 + 0.738i)11-s + (0.188 − 0.326i)12-s + (0.196 − 1.11i)13-s + (0.536 − 0.450i)14-s + (−0.449 − 0.377i)15-s + (−0.109 − 0.620i)16-s + (−0.0872 − 0.0317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18058 + 0.662007i\)
\(L(\frac12)\) \(\approx\) \(1.18058 + 0.662007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (2.75 - 3.37i)T \)
good2 \( 1 + (-1.12 - 0.408i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.394 - 2.23i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.09 + 1.90i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.708 + 4.01i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.359 + 0.130i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (2.27 + 1.91i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (8.05 - 2.93i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.34 + 4.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (-0.544 - 3.08i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.53 + 1.29i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-10.2 + 3.72i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.72 - 4.80i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-12.1 - 4.43i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-4.57 - 3.84i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.986 - 0.358i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.99 - 5.03i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.44 + 13.8i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.33 + 13.2i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.77 + 4.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.13 - 6.46i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (13.5 + 4.94i)T + (74.3 + 62.3i)T^{2} \)
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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

−14.55478430916328096220467733871, −13.38602841555551059253047343296, −12.21229724417569599873875553761, −10.61159312844867641652628572676, −10.10262657278238762645913903605, −8.825936592782905911645967368180, −7.27373812477515737935067774720, −5.63927666387479632362012882044, −4.36540418468001888966904807295, −3.69857689740108606368091438446, 2.15576382944137309663723573755, 3.97370601905861711510539751119, 5.54417638599351482550540662529, 6.93653273742456339438687592600, 8.355987100200929548114168509963, 8.943505830127393336803948599038, 11.35470830002433036230524381507, 11.94437407609862199612464362302, 12.81115410595365817244652513283, 13.68869131365449543273563172421

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