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Properties

Label 2-26-13.9-c7-0-4
Degree 22
Conductor 2626
Sign −0.732+0.680i-0.732 + 0.680i
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  + (4 − 6.92i)2-s + (−39.6 + 68.6i)3-s + (−31.9 − 55.4i)4-s + 132.·5-s + (317. + 549. i)6-s + (−754. − 1.30e3i)7-s − 511.·8-s + (−2.04e3 − 3.54e3i)9-s + (530. − 918. i)10-s + (1.46e3 − 2.53e3i)11-s + 5.07e3·12-s + (−7.91e3 − 240. i)13-s − 1.20e4·14-s + (−5.25e3 + 9.10e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (1.08e4 + 1.88e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.847 + 1.46i)3-s + (−0.249 − 0.433i)4-s + 0.474·5-s + (0.599 + 1.03i)6-s + (−0.831 − 1.43i)7-s − 0.353·8-s + (−0.936 − 1.62i)9-s + (0.167 − 0.290i)10-s + (0.331 − 0.574i)11-s + 0.847·12-s + (−0.999 − 0.0303i)13-s − 1.17·14-s + (−0.402 + 0.696i)15-s + (−0.125 + 0.216i)16-s + (0.536 + 0.929i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((−0.732+0.680i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+7/2) L(s)=((−0.732+0.680i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: −0.732+0.680i-0.732 + 0.680i
Analytic conductor: 8.122018.12201
Root analytic conductor: 2.849912.84991
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ26(9,⋅)\chi_{26} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :7/2), −0.732+0.680i)(2,\ 26,\ (\ :7/2),\ -0.732 + 0.680i)

Particular Values

L(4)L(4) ≈\approx 0.218097−0.555195i0.218097 - 0.555195i
L(12)L(\frac12) ≈\approx 0.218097−0.555195i0.218097 - 0.555195i
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+(−4+6.92i)T 1 + (-4 + 6.92i)T
13 1+(7.91e3+240.i)T 1 + (7.91e3 + 240. i)T
good3 1+(39.6−68.6i)T+(−1.09e3−1.89e3i)T2 1 + (39.6 - 68.6i)T + (-1.09e3 - 1.89e3i)T^{2}
5 1−132.T+7.81e4T2 1 - 132.T + 7.81e4T^{2}
7 1+(754.+1.30e3i)T+(−4.11e5+7.13e5i)T2 1 + (754. + 1.30e3i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+(−1.46e3+2.53e3i)T+(−9.74e6−1.68e7i)T2 1 + (-1.46e3 + 2.53e3i)T + (-9.74e6 - 1.68e7i)T^{2}
17 1+(−1.08e4−1.88e4i)T+(−2.05e8+3.55e8i)T2 1 + (-1.08e4 - 1.88e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(2.67e4+4.63e4i)T+(−4.46e8+7.74e8i)T2 1 + (2.67e4 + 4.63e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+(−1.42e4+2.46e4i)T+(−1.70e9−2.94e9i)T2 1 + (-1.42e4 + 2.46e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(7.63e4−1.32e5i)T+(−8.62e9−1.49e10i)T2 1 + (7.63e4 - 1.32e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1−8.33e4T+2.75e10T2 1 - 8.33e4T + 2.75e10T^{2}
37 1+(5.62e4−9.73e4i)T+(−4.74e10−8.22e10i)T2 1 + (5.62e4 - 9.73e4i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+(5.50e4−9.53e4i)T+(−9.73e10−1.68e11i)T2 1 + (5.50e4 - 9.53e4i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(5.29e4+9.17e4i)T+(−1.35e11+2.35e11i)T2 1 + (5.29e4 + 9.17e4i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1−4.13e5T+5.06e11T2 1 - 4.13e5T + 5.06e11T^{2}
53 1+8.89e5T+1.17e12T2 1 + 8.89e5T + 1.17e12T^{2}
59 1+(1.20e6+2.08e6i)T+(−1.24e12+2.15e12i)T2 1 + (1.20e6 + 2.08e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.19e6+2.06e6i)T+(−1.57e12+2.72e12i)T2 1 + (1.19e6 + 2.06e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−1.84e5+3.20e5i)T+(−3.03e12−5.24e12i)T2 1 + (-1.84e5 + 3.20e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+(−1.59e6−2.76e6i)T+(−4.54e12+7.87e12i)T2 1 + (-1.59e6 - 2.76e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1−5.84e6T+1.10e13T2 1 - 5.84e6T + 1.10e13T^{2}
79 1−1.07e6T+1.92e13T2 1 - 1.07e6T + 1.92e13T^{2}
83 1−6.47e5T+2.71e13T2 1 - 6.47e5T + 2.71e13T^{2}
89 1+(−5.81e6+1.00e7i)T+(−2.21e13−3.83e13i)T2 1 + (-5.81e6 + 1.00e7i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+(3.11e4+5.40e4i)T+(−4.03e13+6.99e13i)T2 1 + (3.11e4 + 5.40e4i)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

−15.46323042147239332232149545198, −14.14185966469543196943693500860, −12.76343221090491441542356669068, −11.08982565283734636784068412085, −10.32530937664371271082009950382, −9.428424844502128801272979713681, −6.41202421035417755654813853444, −4.79274759649410613750469996864, −3.55158826630784322412726203703, −0.28637075526289471709435495819, 2.20771799778827641035146128298, 5.53261499213347158230359298570, 6.34734425840558225407562277420, 7.68362353717483614009415953166, 9.552428130896061108292630343792, 12.07615204928541046212082866356, 12.35069428524569218379174926674, 13.63563467263103993109578490318, 15.06492233485527015483765129699, 16.59742533803894260550626105551

Graph of the ZZ-function along the critical line