Properties

Label 2-95-19.6-c1-0-1
Degree $2$
Conductor $95$
Sign $0.613 - 0.789i$
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  + (−0.454 + 0.381i)2-s + (1.81 + 0.660i)3-s + (−0.286 + 1.62i)4-s + (−0.173 − 0.984i)5-s + (−1.07 + 0.392i)6-s + (−0.530 + 0.918i)7-s + (−1.08 − 1.87i)8-s + (0.555 + 0.466i)9-s + (0.454 + 0.381i)10-s + (−0.0983 − 0.170i)11-s + (−1.58 + 2.75i)12-s + (4.96 − 1.80i)13-s + (−0.109 − 0.620i)14-s + (0.335 − 1.90i)15-s + (−1.88 − 0.686i)16-s + (0.540 − 0.453i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.269i)2-s + (1.04 + 0.381i)3-s + (−0.143 + 0.811i)4-s + (−0.0776 − 0.440i)5-s + (−0.439 + 0.160i)6-s + (−0.200 + 0.347i)7-s + (−0.382 − 0.663i)8-s + (0.185 + 0.155i)9-s + (0.143 + 0.120i)10-s + (−0.0296 − 0.0513i)11-s + (−0.458 + 0.794i)12-s + (1.37 − 0.501i)13-s + (−0.0292 − 0.165i)14-s + (0.0865 − 0.490i)15-s + (−0.471 − 0.171i)16-s + (0.130 − 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953582 + 0.466884i\)
\(L(\frac12)\) \(\approx\) \(0.953582 + 0.466884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (4.24 - 0.983i)T \)
good2 \( 1 + (0.454 - 0.381i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-1.81 - 0.660i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.530 - 0.918i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0983 + 0.170i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.96 + 1.80i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.540 + 0.453i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.15 + 6.52i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (2.59 + 2.17i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.95 - 6.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.09T + 37T^{2} \)
41 \( 1 + (-1.27 - 0.463i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.56 - 8.87i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.69 - 3.09i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.924 - 5.24i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (8.41 - 7.05i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.04 + 11.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.34 + 1.96i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.434 - 2.46i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (6.15 + 2.24i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-11.5 - 4.19i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.01 + 3.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (16.4 - 5.99i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.02 - 0.861i)T + (16.8 - 95.5i)T^{2} \)
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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

−14.18928494918356173724749400195, −13.07873428056999828246946531558, −12.34382972435844517963117958099, −10.79279156110545962389634396497, −9.256703165708221187698820950407, −8.660891763240449719487390137131, −7.909933543149059132879575200152, −6.23522124045540974715672922481, −4.17227855107211888746771007403, −2.99720492964782134570892036342, 1.94753655248776577795791107246, 3.70365300966170389146581462920, 5.80760517315901081263838067498, 7.18412554671242274541798229661, 8.536092598131456556983724291502, 9.329652448014510703960425969367, 10.61063197072456328128400557812, 11.40478551292271112214494956463, 13.22980242579043055281470642298, 13.79952861410700024881185238816

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