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Properties

Label 2-95-19.7-c3-0-5
Degree 22
Conductor 9595
Sign −0.952−0.304i-0.952 - 0.304i
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  + (2.40 + 4.15i)2-s + (1.64 + 2.84i)3-s + (−7.53 + 13.0i)4-s + (2.5 + 4.33i)5-s + (−7.89 + 13.6i)6-s − 11.3·7-s − 33.9·8-s + (8.10 − 14.0i)9-s + (−12.0 + 20.7i)10-s + 22.0·11-s − 49.5·12-s + (3.78 − 6.55i)13-s + (−27.2 − 47.1i)14-s + (−8.21 + 14.2i)15-s + (−21.2 − 36.8i)16-s + (40.9 + 70.9i)17-s + ⋯
L(s)  = 1  + (0.849 + 1.47i)2-s + (0.316 + 0.547i)3-s + (−0.942 + 1.63i)4-s + (0.223 + 0.387i)5-s + (−0.536 + 0.929i)6-s − 0.611·7-s − 1.50·8-s + (0.300 − 0.519i)9-s + (−0.379 + 0.657i)10-s + 0.604·11-s − 1.19·12-s + (0.0807 − 0.139i)13-s + (−0.519 − 0.899i)14-s + (−0.141 + 0.244i)15-s + (−0.332 − 0.576i)16-s + (0.584 + 1.01i)17-s + ⋯

Functional equation

Λ(s)=(95s/2 ΓC(s) L(s)=((−0.952−0.304i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(95s/2 ΓC(s+3/2) L(s)=((−0.952−0.304i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9595    =    5⋅195 \cdot 19
Sign: −0.952−0.304i-0.952 - 0.304i
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), −0.952−0.304i)(2,\ 95,\ (\ :3/2),\ -0.952 - 0.304i)

Particular Values

L(2)L(2) ≈\approx 0.382876+2.45556i0.382876 + 2.45556i
L(12)L(\frac12) ≈\approx 0.382876+2.45556i0.382876 + 2.45556i
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+(82.4+7.97i)T 1 + (82.4 + 7.97i)T
good2 1+(−2.40−4.15i)T+(−4+6.92i)T2 1 + (-2.40 - 4.15i)T + (-4 + 6.92i)T^{2}
3 1+(−1.64−2.84i)T+(−13.5+23.3i)T2 1 + (-1.64 - 2.84i)T + (-13.5 + 23.3i)T^{2}
7 1+11.3T+343T2 1 + 11.3T + 343T^{2}
11 1−22.0T+1.33e3T2 1 - 22.0T + 1.33e3T^{2}
13 1+(−3.78+6.55i)T+(−1.09e3−1.90e3i)T2 1 + (-3.78 + 6.55i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+(−40.9−70.9i)T+(−2.45e3+4.25e3i)T2 1 + (-40.9 - 70.9i)T + (-2.45e3 + 4.25e3i)T^{2}
23 1+(−76.7+132.i)T+(−6.08e3−1.05e4i)T2 1 + (-76.7 + 132. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(−15.8+27.4i)T+(−1.21e4−2.11e4i)T2 1 + (-15.8 + 27.4i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+51.8T+2.97e4T2 1 + 51.8T + 2.97e4T^{2}
37 1−66.6T+5.06e4T2 1 - 66.6T + 5.06e4T^{2}
41 1+(−121.−209.i)T+(−3.44e4+5.96e4i)T2 1 + (-121. - 209. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−143.−248.i)T+(−3.97e4+6.88e4i)T2 1 + (-143. - 248. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−159.+275.i)T+(−5.19e4−8.99e4i)T2 1 + (-159. + 275. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−108.+188.i)T+(−7.44e4−1.28e5i)T2 1 + (-108. + 188. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(275.+477.i)T+(−1.02e5+1.77e5i)T2 1 + (275. + 477. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−91.2+158.i)T+(−1.13e5−1.96e5i)T2 1 + (-91.2 + 158. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−417.+722.i)T+(−1.50e5−2.60e5i)T2 1 + (-417. + 722. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(149.+258.i)T+(−1.78e5+3.09e5i)T2 1 + (149. + 258. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1+(304.+526.i)T+(−1.94e5+3.36e5i)T2 1 + (304. + 526. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(124.+214.i)T+(−2.46e5+4.26e5i)T2 1 + (124. + 214. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+1.19e3T+5.71e5T2 1 + 1.19e3T + 5.71e5T^{2}
89 1+(−343.+594.i)T+(−3.52e5−6.10e5i)T2 1 + (-343. + 594. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(−396.−686.i)T+(−4.56e5+7.90e5i)T2 1 + (-396. - 686. 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

−14.59888546821704214364302711307, −13.13896952868594464549584231073, −12.46062496422427848222145396986, −10.58867007250602309944441057827, −9.343263574422274996911339137031, −8.218763021480389694373672339918, −6.73251918473809794329623912705, −6.13938273649481981279542674558, −4.48201084120380200608372206134, −3.42057592393688813222131978530, 1.28409319467531454437135292061, 2.69406481991469485120454004281, 4.16028475839509612241927240308, 5.54186853618494731996376588878, 7.23004388471215829265933050177, 9.007171966546100244798121371387, 9.988196039852682221366472159196, 11.12361898364166511336399513238, 12.21644711877008451065474090491, 12.94051427235809595285236487820

Graph of the ZZ-function along the critical line