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Properties

Label 2-95-5.4-c1-0-3
Degree $2$
Conductor $95$
Sign $0.871 + 0.490i$
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  − 1.82i·2-s + 2.31i·3-s − 1.32·4-s + (1.94 + 1.09i)5-s + 4.21·6-s − 1.45i·7-s − 1.23i·8-s − 2.35·9-s + (1.99 − 3.55i)10-s − 3.89·11-s − 3.05i·12-s + 3.05i·13-s − 2.64·14-s + (−2.53 + 4.50i)15-s − 4.89·16-s − 3.92i·17-s + ⋯
L(s)  = 1  − 1.28i·2-s + 1.33i·3-s − 0.660·4-s + (0.871 + 0.490i)5-s + 1.72·6-s − 0.548i·7-s − 0.437i·8-s − 0.785·9-s + (0.632 − 1.12i)10-s − 1.17·11-s − 0.883i·12-s + 0.848i·13-s − 0.706·14-s + (−0.655 + 1.16i)15-s − 1.22·16-s − 0.951i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.871 + 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05626 - 0.277012i\)
\(L(\frac12)\) \(\approx\) \(1.05626 - 0.277012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.94 - 1.09i)T \)
19 \( 1 - T \)
good2 \( 1 + 1.82iT - 2T^{2} \)
3 \( 1 - 2.31iT - 3T^{2} \)
7 \( 1 + 1.45iT - 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
13 \( 1 - 3.05iT - 13T^{2} \)
17 \( 1 + 3.92iT - 17T^{2} \)
23 \( 1 + 5.37iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8.43T + 31T^{2} \)
37 \( 1 - 5.95iT - 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 1.45iT - 43T^{2} \)
47 \( 1 + 4.90iT - 47T^{2} \)
53 \( 1 + 4.23iT - 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 9.84iT - 67T^{2} \)
71 \( 1 - 8.64T + 71T^{2} \)
73 \( 1 + 2.43iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 3.05iT - 97T^{2} \)
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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

−13.76152162196642331198529026508, −12.81273552308312570717438258528, −11.25515354277995623663530703661, −10.68068400620436233678923709066, −9.889558497094962574476304690893, −9.207466421927130979777135282763, −7.04633542304467265456813551013, −5.19358768931175998064140747191, −3.86583996974556787142580640504, −2.48444003873614123572081170349, 2.15230438059070685743244395812, 5.49787293209163408559883510113, 5.90925179156571465938361727723, 7.38596925533795779094417833298, 8.036007394564110572714648132899, 9.247546173069956321486330224870, 10.93125477452566170017319745119, 12.61804619216573382055762373068, 13.04424677051916427405246267076, 14.04942844540914308150756089261

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