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Properties

Label 2-95-95.18-c1-0-4
Degree $2$
Conductor $95$
Sign $0.711 + 0.702i$
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  − 2i·4-s + (0.5 − 2.17i)5-s + (−0.679 + 0.679i)7-s + 3i·9-s + 4.35·11-s − 4·16-s + (−5.67 + 5.67i)17-s − 4.35i·19-s + (−4.35 − i)20-s + (6.35 + 6.35i)23-s + (−4.50 − 2.17i)25-s + (1.35 + 1.35i)28-s + (1.14 + 1.82i)35-s + 6·36-s + (−7.03 − 7.03i)43-s − 8.71i·44-s + ⋯
L(s)  = 1  − i·4-s + (0.223 − 0.974i)5-s + (−0.256 + 0.256i)7-s + i·9-s + 1.31·11-s − 16-s + (−1.37 + 1.37i)17-s − 0.999i·19-s + (−0.974 − 0.223i)20-s + (1.32 + 1.32i)23-s + (−0.900 − 0.435i)25-s + (0.256 + 0.256i)28-s + (0.192 + 0.307i)35-s + 36-s + (−1.07 − 1.07i)43-s − 1.31i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941241 - 0.386400i\)
\(L(\frac12)\) \(\approx\) \(0.941241 - 0.386400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 2.17i)T \)
19 \( 1 + 4.35iT \)
good2 \( 1 + 2iT^{2} \)
3 \( 1 - 3iT^{2} \)
7 \( 1 + (0.679 - 0.679i)T - 7iT^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (5.67 - 5.67i)T - 17iT^{2} \)
23 \( 1 + (-6.35 - 6.35i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.03 + 7.03i)T + 43iT^{2} \)
47 \( 1 + (-4.32 + 4.32i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 4.35T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12.0 + 12.0i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3.64 + 3.64i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−13.66380411477519684320127107614, −13.10878433914813782495646729320, −11.62286518404126236888896131871, −10.66223479356514005611971255543, −9.330024964433141136361409933874, −8.712871041969073494609615496972, −6.81740786871571843627761261054, −5.58335680625308499831788154845, −4.45712619263868162916102449913, −1.72354988855230705735800823677, 2.94115436136904689826847043309, 4.14979996239562164668399218937, 6.56026508530815464094768700367, 7.01211270728173269627965594827, 8.722600867732904083858586844803, 9.650133189623906835597430865893, 11.16687314856989593873381931411, 11.92798150806475824763994183989, 13.05542375533788608696513465888, 14.15909926509981515001432653342

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