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Properties

Label 2-26-13.10-c7-0-7
Degree 22
Conductor 2626
Sign −0.978−0.203i-0.978 - 0.203i
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 + (−39.6 − 68.7i)3-s + (31.9 − 55.4i)4-s + 12.5i·5-s + (−549. − 317. i)6-s + (−93.7 − 54.1i)7-s − 511. i·8-s + (−2.05e3 + 3.56e3i)9-s + (50.2 + 87.0i)10-s + (−5.77e3 + 3.33e3i)11-s − 5.07e3·12-s + (5.20e3 − 5.97e3i)13-s − 865.·14-s + (863. − 498. i)15-s + (−2.04e3 − 3.54e3i)16-s + (8.26e3 − 1.43e4i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.848 − 1.46i)3-s + (0.249 − 0.433i)4-s + 0.0449i·5-s + (−1.03 − 0.600i)6-s + (−0.103 − 0.0596i)7-s − 0.353i·8-s + (−0.940 + 1.62i)9-s + (0.0158 + 0.0275i)10-s + (−1.30 + 0.755i)11-s − 0.848·12-s + (0.656 − 0.754i)13-s − 0.0843·14-s + (0.0660 − 0.0381i)15-s + (−0.125 − 0.216i)16-s + (0.408 − 0.706i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((−0.978−0.203i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+7/2) L(s)=((−0.978−0.203i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: −0.978−0.203i-0.978 - 0.203i
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), −0.978−0.203i)(2,\ 26,\ (\ :7/2),\ -0.978 - 0.203i)

Particular Values

L(4)L(4) ≈\approx 0.117079+1.13606i0.117079 + 1.13606i
L(12)L(\frac12) ≈\approx 0.117079+1.13606i0.117079 + 1.13606i
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+(−5.20e3+5.97e3i)T 1 + (-5.20e3 + 5.97e3i)T
good3 1+(39.6+68.7i)T+(−1.09e3+1.89e3i)T2 1 + (39.6 + 68.7i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1−12.5iT−7.81e4T2 1 - 12.5iT - 7.81e4T^{2}
7 1+(93.7+54.1i)T+(4.11e5+7.13e5i)T2 1 + (93.7 + 54.1i)T + (4.11e5 + 7.13e5i)T^{2}
11 1+(5.77e3−3.33e3i)T+(9.74e6−1.68e7i)T2 1 + (5.77e3 - 3.33e3i)T + (9.74e6 - 1.68e7i)T^{2}
17 1+(−8.26e3+1.43e4i)T+(−2.05e8−3.55e8i)T2 1 + (-8.26e3 + 1.43e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(3.05e4+1.76e4i)T+(4.46e8+7.74e8i)T2 1 + (3.05e4 + 1.76e4i)T + (4.46e8 + 7.74e8i)T^{2}
23 1+(1.39e4+2.42e4i)T+(−1.70e9+2.94e9i)T2 1 + (1.39e4 + 2.42e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(1.66e4+2.88e4i)T+(−8.62e9+1.49e10i)T2 1 + (1.66e4 + 2.88e4i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+2.20e5iT−2.75e10T2 1 + 2.20e5iT - 2.75e10T^{2}
37 1+(−1.65e5+9.55e4i)T+(4.74e10−8.22e10i)T2 1 + (-1.65e5 + 9.55e4i)T + (4.74e10 - 8.22e10i)T^{2}
41 1+(7.05e4−4.07e4i)T+(9.73e10−1.68e11i)T2 1 + (7.05e4 - 4.07e4i)T + (9.73e10 - 1.68e11i)T^{2}
43 1+(4.84e5−8.38e5i)T+(−1.35e11−2.35e11i)T2 1 + (4.84e5 - 8.38e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+1.10e6iT−5.06e11T2 1 + 1.10e6iT - 5.06e11T^{2}
53 1−1.60e5T+1.17e12T2 1 - 1.60e5T + 1.17e12T^{2}
59 1+(−1.93e6−1.11e6i)T+(1.24e12+2.15e12i)T2 1 + (-1.93e6 - 1.11e6i)T + (1.24e12 + 2.15e12i)T^{2}
61 1+(9.34e5−1.61e6i)T+(−1.57e12−2.72e12i)T2 1 + (9.34e5 - 1.61e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(−3.10e6+1.79e6i)T+(3.03e12−5.24e12i)T2 1 + (-3.10e6 + 1.79e6i)T + (3.03e12 - 5.24e12i)T^{2}
71 1+(−6.40e5−3.69e5i)T+(4.54e12+7.87e12i)T2 1 + (-6.40e5 - 3.69e5i)T + (4.54e12 + 7.87e12i)T^{2}
73 1+4.24e6iT−1.10e13T2 1 + 4.24e6iT - 1.10e13T^{2}
79 1−2.45e6T+1.92e13T2 1 - 2.45e6T + 1.92e13T^{2}
83 1−5.19e6iT−2.71e13T2 1 - 5.19e6iT - 2.71e13T^{2}
89 1+(−3.73e6+2.15e6i)T+(2.21e13−3.83e13i)T2 1 + (-3.73e6 + 2.15e6i)T + (2.21e13 - 3.83e13i)T^{2}
97 1+(−7.39e6−4.27e6i)T+(4.03e13+6.99e13i)T2 1 + (-7.39e6 - 4.27e6i)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

−15.04385052480766598393517429954, −13.24015562999806584072295180505, −12.91706630222117049417192767867, −11.61386709412650523267148955876, −10.44884752091934324325533139687, −7.85579486230397650621220371352, −6.53321831304394322712130495265, −5.17510533056352246801075894886, −2.37990944217936950273259352152, −0.51590877590071969066773570284, 3.59045083702080530927474020784, 5.02729663381648223485762046152, 6.17694098777086146768695800518, 8.556803565677085355736546742148, 10.33066978805869215065466876957, 11.19960150795027630442843170585, 12.71618915606532455178965506443, 14.33486800689302985194822935378, 15.60081821679508792938281135372, 16.25320499020706533926400700840

Graph of the ZZ-function along the critical line