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Properties

Label 2-95-19.7-c3-0-3
Degree $2$
Conductor $95$
Sign $0.894 - 0.446i$
Analytic cond. $5.60518$
Root an. cond. $2.36752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 2.34i)2-s + (4.30 + 7.44i)3-s + (0.319 − 0.553i)4-s + (2.5 + 4.33i)5-s + (11.6 − 20.2i)6-s + 10.4·7-s − 23.4·8-s + (−23.4 + 40.6i)9-s + (6.78 − 11.7i)10-s + 54.4·11-s + 5.49·12-s + (−40.3 + 69.9i)13-s + (−14.2 − 24.6i)14-s + (−21.5 + 37.2i)15-s + (29.2 + 50.6i)16-s + (9.86 + 17.0i)17-s + ⋯
L(s)  = 1  + (−0.479 − 0.830i)2-s + (0.827 + 1.43i)3-s + (0.0399 − 0.0691i)4-s + (0.223 + 0.387i)5-s + (0.793 − 1.37i)6-s + 0.566·7-s − 1.03·8-s + (−0.870 + 1.50i)9-s + (0.214 − 0.371i)10-s + 1.49·11-s + 0.132·12-s + (−0.861 + 1.49i)13-s + (−0.271 − 0.470i)14-s + (−0.370 + 0.641i)15-s + (0.456 + 0.791i)16-s + (0.140 + 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(5.60518\)
Root analytic conductor: \(2.36752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :3/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69302 + 0.399436i\)
\(L(\frac12)\) \(\approx\) \(1.69302 + 0.399436i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.5 - 4.33i)T \)
19 \( 1 + (-64.5 + 51.8i)T \)
good2 \( 1 + (1.35 + 2.34i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.30 - 7.44i)T + (-13.5 + 23.3i)T^{2} \)
7 \( 1 - 10.4T + 343T^{2} \)
11 \( 1 - 54.4T + 1.33e3T^{2} \)
13 \( 1 + (40.3 - 69.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-9.86 - 17.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-64.9 + 112. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (13.6 - 23.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 9.90T + 2.97e4T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 + (258. + 447. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-128. - 223. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-66.2 + 114. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-199. + 345. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (298. + 516. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-183. + 317. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (83.7 - 145. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (331. + 574. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (66.6 + 115. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-1.95 - 3.38i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 154.T + 5.71e5T^{2} \)
89 \( 1 + (-534. + 924. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (367. + 637. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.20657418628434972942945910264, −11.98435044102541288796648008897, −11.19276338294576093145995085038, −10.19785966348505928529637078197, −9.314615242581867204492201155002, −8.823685079011574403106544652876, −6.76970009392086692688620281803, −4.82755907158703259075951222818, −3.49718242765255502892025739421, −2.01674595870786965299758792455, 1.26095082385085948443012952105, 3.08742762738865644901371500052, 5.68127566439890424041718908071, 7.01670505813238605077203803778, 7.72620896402145571481257257037, 8.566126903806136066513746362588, 9.569621573426856193623630414043, 11.80389779720182208261989103368, 12.38384153370508708147514091265, 13.54128332115120139229657462633

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