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Properties

Label 2-26-13.10-c7-0-6
Degree 22
Conductor 2626
Sign −0.752−0.659i-0.752 - 0.659i
Analytic cond. 8.122018.12201
Root an. cond. 2.849912.84991
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.92 + 4i)2-s + (−34.8 − 60.3i)3-s + (31.9 − 55.4i)4-s − 226. i·5-s + (483. + 278. i)6-s + (−154. − 89.2i)7-s + 511. i·8-s + (−1.33e3 + 2.31e3i)9-s + (907. + 1.57e3i)10-s + (−762. + 440. i)11-s − 4.46e3·12-s + (−7.46e3 + 2.64e3i)13-s + 1.42e3·14-s + (−1.36e4 + 7.90e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (−5.09e3 + 8.81e3i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.745 − 1.29i)3-s + (0.249 − 0.433i)4-s − 0.811i·5-s + (0.912 + 0.527i)6-s + (−0.170 − 0.0983i)7-s + 0.353i·8-s + (−0.611 + 1.05i)9-s + (0.286 + 0.496i)10-s + (−0.172 + 0.0996i)11-s − 0.745·12-s + (−0.942 + 0.333i)13-s + 0.139·14-s + (−1.04 + 0.604i)15-s + (−0.125 − 0.216i)16-s + (−0.251 + 0.435i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((−0.752−0.659i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.659i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+7/2) L(s)=((−0.752−0.659i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: −0.752−0.659i-0.752 - 0.659i
Analytic conductor: 8.122018.12201
Root analytic conductor: 2.849912.84991
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ26(23,⋅)\chi_{26} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :7/2), −0.752−0.659i)(2,\ 26,\ (\ :7/2),\ -0.752 - 0.659i)

Particular Values

L(4)L(4) ≈\approx 0.0610159+0.162211i0.0610159 + 0.162211i
L(12)L(\frac12) ≈\approx 0.0610159+0.162211i0.0610159 + 0.162211i
L(92)L(\frac{9}{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)
bad2 1+(6.92−4i)T 1 + (6.92 - 4i)T
13 1+(7.46e3−2.64e3i)T 1 + (7.46e3 - 2.64e3i)T
good3 1+(34.8+60.3i)T+(−1.09e3+1.89e3i)T2 1 + (34.8 + 60.3i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+226.iT−7.81e4T2 1 + 226. iT - 7.81e4T^{2}
7 1+(154.+89.2i)T+(4.11e5+7.13e5i)T2 1 + (154. + 89.2i)T + (4.11e5 + 7.13e5i)T^{2}
11 1+(762.−440.i)T+(9.74e6−1.68e7i)T2 1 + (762. - 440. i)T + (9.74e6 - 1.68e7i)T^{2}
17 1+(5.09e3−8.81e3i)T+(−2.05e8−3.55e8i)T2 1 + (5.09e3 - 8.81e3i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(−9.79e3−5.65e3i)T+(4.46e8+7.74e8i)T2 1 + (-9.79e3 - 5.65e3i)T + (4.46e8 + 7.74e8i)T^{2}
23 1+(−4.60e4−7.97e4i)T+(−1.70e9+2.94e9i)T2 1 + (-4.60e4 - 7.97e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(7.93e4+1.37e5i)T+(−8.62e9+1.49e10i)T2 1 + (7.93e4 + 1.37e5i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+1.95e5iT−2.75e10T2 1 + 1.95e5iT - 2.75e10T^{2}
37 1+(4.47e5−2.58e5i)T+(4.74e10−8.22e10i)T2 1 + (4.47e5 - 2.58e5i)T + (4.74e10 - 8.22e10i)T^{2}
41 1+(5.63e5−3.25e5i)T+(9.73e10−1.68e11i)T2 1 + (5.63e5 - 3.25e5i)T + (9.73e10 - 1.68e11i)T^{2}
43 1+(−3.46e5+5.99e5i)T+(−1.35e11−2.35e11i)T2 1 + (-3.46e5 + 5.99e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1−5.23e5iT−5.06e11T2 1 - 5.23e5iT - 5.06e11T^{2}
53 1+1.07e6T+1.17e12T2 1 + 1.07e6T + 1.17e12T^{2}
59 1+(1.80e6+1.04e6i)T+(1.24e12+2.15e12i)T2 1 + (1.80e6 + 1.04e6i)T + (1.24e12 + 2.15e12i)T^{2}
61 1+(−7.43e5+1.28e6i)T+(−1.57e12−2.72e12i)T2 1 + (-7.43e5 + 1.28e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(6.86e5−3.96e5i)T+(3.03e12−5.24e12i)T2 1 + (6.86e5 - 3.96e5i)T + (3.03e12 - 5.24e12i)T^{2}
71 1+(−4.33e6−2.50e6i)T+(4.54e12+7.87e12i)T2 1 + (-4.33e6 - 2.50e6i)T + (4.54e12 + 7.87e12i)T^{2}
73 1+2.74e6iT−1.10e13T2 1 + 2.74e6iT - 1.10e13T^{2}
79 1+2.95e6T+1.92e13T2 1 + 2.95e6T + 1.92e13T^{2}
83 1−6.23e6iT−2.71e13T2 1 - 6.23e6iT - 2.71e13T^{2}
89 1+(2.70e6−1.56e6i)T+(2.21e13−3.83e13i)T2 1 + (2.70e6 - 1.56e6i)T + (2.21e13 - 3.83e13i)T^{2}
97 1+(5.04e6+2.91e6i)T+(4.03e13+6.99e13i)T2 1 + (5.04e6 + 2.91e6i)T + (4.03e13 + 6.99e13i)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

−15.38648542523723521865256970696, −13.56363594343123785023814252528, −12.49886737943253041545341308862, −11.41145100506111394543258473487, −9.570097910191547616892413640493, −7.935853897189349539682281824661, −6.79211665126987161518610494449, −5.31276606091457072559279677294, −1.61087922382475485658296601917, −0.11733695125166515010796978340, 3.06780212944402328204794272954, 5.01282119281704260697295693677, 6.96640931407194190411757032301, 9.093649503693623975723519871512, 10.37056072985871347926515079012, 10.94933018051188361241312957940, 12.38469949785675697195086672794, 14.53123703930002615028244001058, 15.67605501874260052907107924489, 16.65241944727704893452636880362

Graph of the ZZ-function along the critical line