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Properties

Label 2-115-23.4-c3-0-0
Degree 22
Conductor 115115
Sign −0.205−0.978i-0.205 - 0.978i
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
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  + (−1.13 − 2.48i)2-s + (−4.84 + 1.42i)3-s + (0.356 − 0.411i)4-s + (4.20 + 2.70i)5-s + (9.02 + 10.4i)6-s + (−3.26 − 22.7i)7-s + (−22.3 − 6.57i)8-s + (−1.29 + 0.830i)9-s + (1.94 − 13.5i)10-s + (−19.7 + 43.1i)11-s + (−1.14 + 2.50i)12-s + (3.63 − 25.3i)13-s + (−52.7 + 33.8i)14-s + (−24.2 − 7.10i)15-s + (8.44 + 58.7i)16-s + (56.7 + 65.5i)17-s + ⋯
L(s)  = 1  + (−0.401 − 0.878i)2-s + (−0.931 + 0.273i)3-s + (0.0445 − 0.0514i)4-s + (0.376 + 0.241i)5-s + (0.613 + 0.708i)6-s + (−0.176 − 1.22i)7-s + (−0.989 − 0.290i)8-s + (−0.0478 + 0.0307i)9-s + (0.0614 − 0.427i)10-s + (−0.540 + 1.18i)11-s + (−0.0274 + 0.0601i)12-s + (0.0776 − 0.539i)13-s + (−1.00 + 0.647i)14-s + (−0.416 − 0.122i)15-s + (0.131 + 0.917i)16-s + (0.809 + 0.934i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.205−0.978i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((−0.205−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.205−0.978i-0.205 - 0.978i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(96,⋅)\chi_{115} (96, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), −0.205−0.978i)(2,\ 115,\ (\ :3/2),\ -0.205 - 0.978i)

Particular Values

L(2)L(2) ≈\approx 0.0612224+0.0753821i0.0612224 + 0.0753821i
L(12)L(\frac12) ≈\approx 0.0612224+0.0753821i0.0612224 + 0.0753821i
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+(−4.20−2.70i)T 1 + (-4.20 - 2.70i)T
23 1+(87.1−67.6i)T 1 + (87.1 - 67.6i)T
good2 1+(1.13+2.48i)T+(−5.23+6.04i)T2 1 + (1.13 + 2.48i)T + (-5.23 + 6.04i)T^{2}
3 1+(4.84−1.42i)T+(22.7−14.5i)T2 1 + (4.84 - 1.42i)T + (22.7 - 14.5i)T^{2}
7 1+(3.26+22.7i)T+(−329.+96.6i)T2 1 + (3.26 + 22.7i)T + (-329. + 96.6i)T^{2}
11 1+(19.7−43.1i)T+(−871.−1.00e3i)T2 1 + (19.7 - 43.1i)T + (-871. - 1.00e3i)T^{2}
13 1+(−3.63+25.3i)T+(−2.10e3−618.i)T2 1 + (-3.63 + 25.3i)T + (-2.10e3 - 618. i)T^{2}
17 1+(−56.7−65.5i)T+(−699.+4.86e3i)T2 1 + (-56.7 - 65.5i)T + (-699. + 4.86e3i)T^{2}
19 1+(76.3−88.1i)T+(−976.−6.78e3i)T2 1 + (76.3 - 88.1i)T + (-976. - 6.78e3i)T^{2}
29 1+(32.8+37.8i)T+(−3.47e3+2.41e4i)T2 1 + (32.8 + 37.8i)T + (-3.47e3 + 2.41e4i)T^{2}
31 1+(149.+43.8i)T+(2.50e4+1.61e4i)T2 1 + (149. + 43.8i)T + (2.50e4 + 1.61e4i)T^{2}
37 1+(229.−147.i)T+(2.10e4−4.60e4i)T2 1 + (229. - 147. i)T + (2.10e4 - 4.60e4i)T^{2}
41 1+(−20.7−13.3i)T+(2.86e4+6.26e4i)T2 1 + (-20.7 - 13.3i)T + (2.86e4 + 6.26e4i)T^{2}
43 1+(287.−84.3i)T+(6.68e4−4.29e4i)T2 1 + (287. - 84.3i)T + (6.68e4 - 4.29e4i)T^{2}
47 1−249.T+1.03e5T2 1 - 249.T + 1.03e5T^{2}
53 1+(105.+737.i)T+(−1.42e5+4.19e4i)T2 1 + (105. + 737. i)T + (-1.42e5 + 4.19e4i)T^{2}
59 1+(−56.0+390.i)T+(−1.97e5−5.78e4i)T2 1 + (-56.0 + 390. i)T + (-1.97e5 - 5.78e4i)T^{2}
61 1+(−37.5−11.0i)T+(1.90e5+1.22e5i)T2 1 + (-37.5 - 11.0i)T + (1.90e5 + 1.22e5i)T^{2}
67 1+(330.+724.i)T+(−1.96e5+2.27e5i)T2 1 + (330. + 724. i)T + (-1.96e5 + 2.27e5i)T^{2}
71 1+(−237.−520.i)T+(−2.34e5+2.70e5i)T2 1 + (-237. - 520. i)T + (-2.34e5 + 2.70e5i)T^{2}
73 1+(−128.+148.i)T+(−5.53e4−3.85e5i)T2 1 + (-128. + 148. i)T + (-5.53e4 - 3.85e5i)T^{2}
79 1+(−42.7+297.i)T+(−4.73e5−1.38e5i)T2 1 + (-42.7 + 297. i)T + (-4.73e5 - 1.38e5i)T^{2}
83 1+(972.−625.i)T+(2.37e5−5.20e5i)T2 1 + (972. - 625. i)T + (2.37e5 - 5.20e5i)T^{2}
89 1+(−871.+255.i)T+(5.93e5−3.81e5i)T2 1 + (-871. + 255. i)T + (5.93e5 - 3.81e5i)T^{2}
97 1+(1.38e3+888.i)T+(3.79e5+8.30e5i)T2 1 + (1.38e3 + 888. i)T + (3.79e5 + 8.30e5i)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

−12.98815512590282284759346813859, −12.14338529289985149005935831401, −10.98122573978632192925051475723, −10.19776431928860644644633726514, −10.00984876878327785951734665113, −7.983467510118603291249100209478, −6.53521211749335315041219369711, −5.44507939744148523173026819275, −3.72844039711461989523501347478, −1.77926098007103352911081250686, 0.06224546697333398272858682455, 2.75163514414393636770310066855, 5.42741216162228261164115944942, 5.95627744257267731536096155624, 7.02014625638075030019010475037, 8.557036061629525782373087075878, 9.108782134747149798455222901074, 10.88186382970625015870171293400, 11.88672645362731678727856666429, 12.50508135993430544720288128738

Graph of the ZZ-function along the critical line