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Properties

Label 2-95-19.7-c3-0-3
Degree 22
Conductor 9595
Sign 0.894−0.446i0.894 - 0.446i
Analytic cond. 5.605185.60518
Root an. cond. 2.367522.36752
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(95s/2 ΓC(s) L(s)=((0.894−0.446i) Λ‾(4−s)\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}
Λ(s)=(95s/2 ΓC(s+3/2) L(s)=((0.894−0.446i) Λ‾(1−s)\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: 22
Conductor: 9595    =    5⋅195 \cdot 19
Sign: 0.894−0.446i0.894 - 0.446i
Analytic conductor: 5.605185.60518
Root analytic conductor: 2.367522.36752
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ95(26,⋅)\chi_{95} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 95, ( :3/2), 0.894−0.446i)(2,\ 95,\ (\ :3/2),\ 0.894 - 0.446i)

Particular Values

L(2)L(2) ≈\approx 1.69302+0.399436i1.69302 + 0.399436i
L(12)L(\frac12) ≈\approx 1.69302+0.399436i1.69302 + 0.399436i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−2.5−4.33i)T 1 + (-2.5 - 4.33i)T
19 1+(−64.5+51.8i)T 1 + (-64.5 + 51.8i)T
good2 1+(1.35+2.34i)T+(−4+6.92i)T2 1 + (1.35 + 2.34i)T + (-4 + 6.92i)T^{2}
3 1+(−4.30−7.44i)T+(−13.5+23.3i)T2 1 + (-4.30 - 7.44i)T + (-13.5 + 23.3i)T^{2}
7 1−10.4T+343T2 1 - 10.4T + 343T^{2}
11 1−54.4T+1.33e3T2 1 - 54.4T + 1.33e3T^{2}
13 1+(40.3−69.9i)T+(−1.09e3−1.90e3i)T2 1 + (40.3 - 69.9i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+(−9.86−17.0i)T+(−2.45e3+4.25e3i)T2 1 + (-9.86 - 17.0i)T + (-2.45e3 + 4.25e3i)T^{2}
23 1+(−64.9+112.i)T+(−6.08e3−1.05e4i)T2 1 + (-64.9 + 112. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(13.6−23.6i)T+(−1.21e4−2.11e4i)T2 1 + (13.6 - 23.6i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1−9.90T+2.97e4T2 1 - 9.90T + 2.97e4T^{2}
37 1+217.T+5.06e4T2 1 + 217.T + 5.06e4T^{2}
41 1+(258.+447.i)T+(−3.44e4+5.96e4i)T2 1 + (258. + 447. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−128.−223.i)T+(−3.97e4+6.88e4i)T2 1 + (-128. - 223. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−66.2+114.i)T+(−5.19e4−8.99e4i)T2 1 + (-66.2 + 114. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−199.+345.i)T+(−7.44e4−1.28e5i)T2 1 + (-199. + 345. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(298.+516.i)T+(−1.02e5+1.77e5i)T2 1 + (298. + 516. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−183.+317.i)T+(−1.13e5−1.96e5i)T2 1 + (-183. + 317. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(83.7−145.i)T+(−1.50e5−2.60e5i)T2 1 + (83.7 - 145. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(331.+574.i)T+(−1.78e5+3.09e5i)T2 1 + (331. + 574. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1+(66.6+115.i)T+(−1.94e5+3.36e5i)T2 1 + (66.6 + 115. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−1.95−3.38i)T+(−2.46e5+4.26e5i)T2 1 + (-1.95 - 3.38i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−154.T+5.71e5T2 1 - 154.T + 5.71e5T^{2}
89 1+(−534.+924.i)T+(−3.52e5−6.10e5i)T2 1 + (-534. + 924. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(367.+637.i)T+(−4.56e5+7.90e5i)T2 1 + (367. + 637. i)T + (-4.56e5 + 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line