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Properties

Label 2-74-37.10-c7-0-11
Degree 22
Conductor 7474
Sign 0.0760+0.997i0.0760 + 0.997i
Analytic cond. 23.116423.1164
Root an. cond. 4.807964.80796
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 + (8.02 + 13.9i)3-s + (−31.9 − 55.4i)4-s + (−60.7 − 105. i)5-s + 128.·6-s + (462. + 801. i)7-s − 511.·8-s + (964. − 1.67e3i)9-s − 972.·10-s + 1.63e3·11-s + (513. − 890. i)12-s + (1.56e3 + 2.71e3i)13-s + 7.40e3·14-s + (975. − 1.69e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (8.23e3 − 1.42e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.171 + 0.297i)3-s + (−0.249 − 0.433i)4-s + (−0.217 − 0.376i)5-s + 0.242·6-s + (0.509 + 0.883i)7-s − 0.353·8-s + (0.441 − 0.763i)9-s − 0.307·10-s + 0.369·11-s + (0.0858 − 0.148i)12-s + (0.197 + 0.342i)13-s + 0.721·14-s + (0.0746 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.406 − 0.704i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.0760+0.997i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0760 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((0.0760+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0760 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.0760+0.997i0.0760 + 0.997i
Analytic conductor: 23.116423.1164
Root analytic conductor: 4.807964.80796
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ74(47,⋅)\chi_{74} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :7/2), 0.0760+0.997i)(2,\ 74,\ (\ :7/2),\ 0.0760 + 0.997i)

Particular Values

L(4)L(4) ≈\approx 1.79071−1.65932i1.79071 - 1.65932i
L(12)L(\frac12) ≈\approx 1.79071−1.65932i1.79071 - 1.65932i
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
37 1+(−2.18e4+3.07e5i)T 1 + (-2.18e4 + 3.07e5i)T
good3 1+(−8.02−13.9i)T+(−1.09e3+1.89e3i)T2 1 + (-8.02 - 13.9i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+(60.7+105.i)T+(−3.90e4+6.76e4i)T2 1 + (60.7 + 105. i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(−462.−801.i)T+(−4.11e5+7.13e5i)T2 1 + (-462. - 801. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1−1.63e3T+1.94e7T2 1 - 1.63e3T + 1.94e7T^{2}
13 1+(−1.56e3−2.71e3i)T+(−3.13e7+5.43e7i)T2 1 + (-1.56e3 - 2.71e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+(−8.23e3+1.42e4i)T+(−2.05e8−3.55e8i)T2 1 + (-8.23e3 + 1.42e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(2.54e4+4.41e4i)T+(−4.46e8+7.74e8i)T2 1 + (2.54e4 + 4.41e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1−6.04e4T+3.40e9T2 1 - 6.04e4T + 3.40e9T^{2}
29 1+9.97e4T+1.72e10T2 1 + 9.97e4T + 1.72e10T^{2}
31 1−1.44e5T+2.75e10T2 1 - 1.44e5T + 2.75e10T^{2}
41 1+(4.04e5+7.00e5i)T+(−9.73e10+1.68e11i)T2 1 + (4.04e5 + 7.00e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+7.00e4T+2.71e11T2 1 + 7.00e4T + 2.71e11T^{2}
47 1−7.04e5T+5.06e11T2 1 - 7.04e5T + 5.06e11T^{2}
53 1+(5.73e5−9.93e5i)T+(−5.87e11−1.01e12i)T2 1 + (5.73e5 - 9.93e5i)T + (-5.87e11 - 1.01e12i)T^{2}
59 1+(4.76e5−8.25e5i)T+(−1.24e12−2.15e12i)T2 1 + (4.76e5 - 8.25e5i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−6.70e5−1.16e6i)T+(−1.57e12+2.72e12i)T2 1 + (-6.70e5 - 1.16e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(1.01e6+1.75e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.01e6 + 1.75e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(−2.52e6−4.37e6i)T+(−4.54e12+7.87e12i)T2 1 + (-2.52e6 - 4.37e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1−4.62e6T+1.10e13T2 1 - 4.62e6T + 1.10e13T^{2}
79 1+(−2.53e6−4.39e6i)T+(−9.60e12+1.66e13i)T2 1 + (-2.53e6 - 4.39e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(6.51e5−1.12e6i)T+(−1.35e13−2.35e13i)T2 1 + (6.51e5 - 1.12e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+(6.04e6−1.04e7i)T+(−2.21e13−3.83e13i)T2 1 + (6.04e6 - 1.04e7i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1−3.50e6T+8.07e13T2 1 - 3.50e6T + 8.07e13T^{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

−12.62296287078764406833572777515, −11.88914477434364765422311055413, −10.83540009075327999076832236972, −9.312799961865092770033109283688, −8.740067906462451696788841134894, −6.80575391268906746177841284818, −5.19548401117202969505329463247, −4.07944344364412496133671769682, −2.51286261732955357907021711219, −0.822852523879441203252774627729, 1.45448838406535358379585285857, 3.52313703828169779567885446959, 4.80337884519493292068749385033, 6.41961363676768307658843964211, 7.56984192011830577051082987372, 8.302042095644162646701019478083, 10.15642305847020376768311871654, 11.14195540636219631439488757461, 12.63053086042851861937344700366, 13.50074370807666433401864032363

Graph of the ZZ-function along the critical line