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Properties

Label 2-126-7.2-c5-0-1
Degree 22
Conductor 126126
Sign 0.249−0.968i0.249 - 0.968i
Analytic cond. 20.208320.2083
Root an. cond. 4.495374.49537
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (11.2 − 19.4i)5-s + (−96.4 + 86.5i)7-s − 63.9·8-s + (−44.9 − 77.9i)10-s + (170. + 294. i)11-s − 728.·13-s + (106. + 507. i)14-s + (−128 + 221. i)16-s + (404. + 701. i)17-s + (−513. + 888. i)19-s − 359.·20-s + 1.36e3·22-s + (711. − 1.23e3i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.201 − 0.348i)5-s + (−0.744 + 0.667i)7-s − 0.353·8-s + (−0.142 − 0.246i)10-s + (0.424 + 0.734i)11-s − 1.19·13-s + (0.145 + 0.691i)14-s + (−0.125 + 0.216i)16-s + (0.339 + 0.588i)17-s + (−0.326 + 0.564i)19-s − 0.201·20-s + 0.599·22-s + (0.280 − 0.485i)23-s + ⋯

Functional equation

Λ(s)=(126s/2 ΓC(s) L(s)=((0.249−0.968i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(126s/2 ΓC(s+5/2) L(s)=((0.249−0.968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 126126    =    2⋅32⋅72 \cdot 3^{2} \cdot 7
Sign: 0.249−0.968i0.249 - 0.968i
Analytic conductor: 20.208320.2083
Root analytic conductor: 4.495374.49537
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ126(37,⋅)\chi_{126} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 126, ( :5/2), 0.249−0.968i)(2,\ 126,\ (\ :5/2),\ 0.249 - 0.968i)

Particular Values

L(3)L(3) ≈\approx 1.0601219841.060121984
L(12)L(\frac12) ≈\approx 1.0601219841.060121984
L(72)L(\frac{7}{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+(−2+3.46i)T 1 + (-2 + 3.46i)T
3 1 1
7 1+(96.4−86.5i)T 1 + (96.4 - 86.5i)T
good5 1+(−11.2+19.4i)T+(−1.56e3−2.70e3i)T2 1 + (-11.2 + 19.4i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−170.−294.i)T+(−8.05e4+1.39e5i)T2 1 + (-170. - 294. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+728.T+3.71e5T2 1 + 728.T + 3.71e5T^{2}
17 1+(−404.−701.i)T+(−7.09e5+1.22e6i)T2 1 + (-404. - 701. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(513.−888.i)T+(−1.23e6−2.14e6i)T2 1 + (513. - 888. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−711.+1.23e3i)T+(−3.21e6−5.57e6i)T2 1 + (-711. + 1.23e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+5.21e3T+2.05e7T2 1 + 5.21e3T + 2.05e7T^{2}
31 1+(−3.51e3−6.09e3i)T+(−1.43e7+2.47e7i)T2 1 + (-3.51e3 - 6.09e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(6.39e3−1.10e4i)T+(−3.46e7−6.00e7i)T2 1 + (6.39e3 - 1.10e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+1.17e3T+1.15e8T2 1 + 1.17e3T + 1.15e8T^{2}
43 1−3.66e3T+1.47e8T2 1 - 3.66e3T + 1.47e8T^{2}
47 1+(−4.65e3+8.06e3i)T+(−1.14e8−1.98e8i)T2 1 + (-4.65e3 + 8.06e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.78e4−3.08e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.78e4 - 3.08e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.51e4+2.63e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.51e4 + 2.63e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(1.60e4−2.78e4i)T+(−4.22e8−7.31e8i)T2 1 + (1.60e4 - 2.78e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(1.06e4+1.84e4i)T+(−6.75e8+1.16e9i)T2 1 + (1.06e4 + 1.84e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+6.11e4T+1.80e9T2 1 + 6.11e4T + 1.80e9T^{2}
73 1+(2.06e4+3.57e4i)T+(−1.03e9+1.79e9i)T2 1 + (2.06e4 + 3.57e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−1.75e4+3.03e4i)T+(−1.53e9−2.66e9i)T2 1 + (-1.75e4 + 3.03e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+8.61e4T+3.93e9T2 1 + 8.61e4T + 3.93e9T^{2}
89 1+(−3.89e4+6.75e4i)T+(−2.79e9−4.83e9i)T2 1 + (-3.89e4 + 6.75e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.61e5T+8.58e9T2 1 + 1.61e5T + 8.58e9T^{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

−12.38397056920781184405751426184, −12.10744426207410414675035052999, −10.48792031285069340291983896714, −9.652441778762173953483579197447, −8.740659253089825151546420509926, −7.06833211016014294132596465115, −5.76361348712872961936788925217, −4.59976368735826400567093840977, −3.07636583610418081012046819235, −1.68054151491906424939103888895, 0.32654658014322787206962656965, 2.79605513249982321686827325637, 4.12457067199298598192794404726, 5.59202045253676122459458449674, 6.76495015722998805396506992118, 7.54755158398408503166965863555, 9.069879192122574898495856516542, 10.01083363831106342268891736571, 11.25742572585084114353443442387, 12.44960581179986635913304281894

Graph of the ZZ-function along the critical line