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Properties

Label 2-26-13.9-c7-0-5
Degree 22
Conductor 2626
Sign −0.236+0.971i-0.236 + 0.971i
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 + (35.5 − 61.6i)3-s + (−31.9 − 55.4i)4-s + 523.·5-s + (−284. − 492. i)6-s + (208. + 360. i)7-s − 511.·8-s + (−1.43e3 − 2.49e3i)9-s + (2.09e3 − 3.62e3i)10-s + (−1.71e3 + 2.96e3i)11-s − 4.55e3·12-s + (−6.74e3 + 4.15e3i)13-s + 3.33e3·14-s + (1.86e4 − 3.22e4i)15-s + (−2.04e3 + 3.54e3i)16-s + (3.26e3 + 5.66e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.760 − 1.31i)3-s + (−0.249 − 0.433i)4-s + 1.87·5-s + (−0.537 − 0.931i)6-s + (0.229 + 0.397i)7-s − 0.353·8-s + (−0.657 − 1.13i)9-s + (0.662 − 1.14i)10-s + (−0.387 + 0.671i)11-s − 0.760·12-s + (−0.851 + 0.525i)13-s + 0.324·14-s + (1.42 − 2.46i)15-s + (−0.125 + 0.216i)16-s + (0.161 + 0.279i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((−0.236+0.971i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+7/2) L(s)=((−0.236+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: −0.236+0.971i-0.236 + 0.971i
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), −0.236+0.971i)(2,\ 26,\ (\ :7/2),\ -0.236 + 0.971i)

Particular Values

L(4)L(4) ≈\approx 1.86555−2.37452i1.86555 - 2.37452i
L(12)L(\frac12) ≈\approx 1.86555−2.37452i1.86555 - 2.37452i
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+(6.74e3−4.15e3i)T 1 + (6.74e3 - 4.15e3i)T
good3 1+(−35.5+61.6i)T+(−1.09e3−1.89e3i)T2 1 + (-35.5 + 61.6i)T + (-1.09e3 - 1.89e3i)T^{2}
5 1−523.T+7.81e4T2 1 - 523.T + 7.81e4T^{2}
7 1+(−208.−360.i)T+(−4.11e5+7.13e5i)T2 1 + (-208. - 360. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+(1.71e3−2.96e3i)T+(−9.74e6−1.68e7i)T2 1 + (1.71e3 - 2.96e3i)T + (-9.74e6 - 1.68e7i)T^{2}
17 1+(−3.26e3−5.66e3i)T+(−2.05e8+3.55e8i)T2 1 + (-3.26e3 - 5.66e3i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(1.39e4+2.42e4i)T+(−4.46e8+7.74e8i)T2 1 + (1.39e4 + 2.42e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+(5.30e4−9.19e4i)T+(−1.70e9−2.94e9i)T2 1 + (5.30e4 - 9.19e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(3.42e4−5.93e4i)T+(−8.62e9−1.49e10i)T2 1 + (3.42e4 - 5.93e4i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1−5.15e4T+2.75e10T2 1 - 5.15e4T + 2.75e10T^{2}
37 1+(2.40e4−4.16e4i)T+(−4.74e10−8.22e10i)T2 1 + (2.40e4 - 4.16e4i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+(−3.01e5+5.21e5i)T+(−9.73e10−1.68e11i)T2 1 + (-3.01e5 + 5.21e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(4.58e5+7.93e5i)T+(−1.35e11+2.35e11i)T2 1 + (4.58e5 + 7.93e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+3.26e5T+5.06e11T2 1 + 3.26e5T + 5.06e11T^{2}
53 1−9.34e5T+1.17e12T2 1 - 9.34e5T + 1.17e12T^{2}
59 1+(5.87e5+1.01e6i)T+(−1.24e12+2.15e12i)T2 1 + (5.87e5 + 1.01e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(−1.44e6−2.50e6i)T+(−1.57e12+2.72e12i)T2 1 + (-1.44e6 - 2.50e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(1.59e5−2.75e5i)T+(−3.03e12−5.24e12i)T2 1 + (1.59e5 - 2.75e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+(6.41e5+1.11e6i)T+(−4.54e12+7.87e12i)T2 1 + (6.41e5 + 1.11e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1+1.67e6T+1.10e13T2 1 + 1.67e6T + 1.10e13T^{2}
79 1+8.09e6T+1.92e13T2 1 + 8.09e6T + 1.92e13T^{2}
83 1−5.57e6T+2.71e13T2 1 - 5.57e6T + 2.71e13T^{2}
89 1+(3.93e6−6.82e6i)T+(−2.21e13−3.83e13i)T2 1 + (3.93e6 - 6.82e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+(−3.40e6−5.89e6i)T+(−4.03e13+6.99e13i)T2 1 + (-3.40e6 - 5.89e6i)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

−14.84676666327333063445700580191, −13.83470799088299567448309680381, −13.18770264774338598207825048793, −12.12843413966603210069818228028, −10.08497043803827190675439014939, −8.942488827783357871327831757949, −7.00867061791101382846931839147, −5.46484886820953479772836985539, −2.38291324254866173764632379237, −1.74495065676387056838552167449, 2.71018029187727279853891285228, 4.67975183157922891483601731651, 6.01761739580206786286193250518, 8.311122336715776955996242379328, 9.695427781029161976108185641649, 10.40711909285565494612530268500, 13.01596632490984819616057486234, 14.21201292946650620119916019944, 14.65132698796591748108293531572, 16.25029011579572675279617308888

Graph of the ZZ-function along the critical line