↑

Properties

Label 2-74-37.10-c7-0-13
Degree 22
Conductor 7474
Sign 0.476+0.879i0.476 + 0.879i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (27.4 + 47.5i)3-s + (−31.9 − 55.4i)4-s + (8.21 + 14.2i)5-s + 439.·6-s + (−392. − 679. i)7-s − 511.·8-s + (−412. + 713. i)9-s + 131.·10-s + 6.02e3·11-s + (1.75e3 − 3.04e3i)12-s + (−6.90e3 − 1.19e4i)13-s − 6.28e3·14-s + (−450. + 781. i)15-s + (−2.04e3 + 3.54e3i)16-s + (6.83e3 − 1.18e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.586 + 1.01i)3-s + (−0.249 − 0.433i)4-s + (0.0294 + 0.0509i)5-s + 0.829·6-s + (−0.432 − 0.749i)7-s − 0.353·8-s + (−0.188 + 0.326i)9-s + 0.0415·10-s + 1.36·11-s + (0.293 − 0.508i)12-s + (−0.872 − 1.51i)13-s − 0.611·14-s + (−0.0345 + 0.0597i)15-s + (−0.125 + 0.216i)16-s + (0.337 − 0.584i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.476+0.879i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((0.476+0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.476+0.879i0.476 + 0.879i
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), 0.476+0.879i)(2,\ 74,\ (\ :7/2),\ 0.476 + 0.879i)

Particular Values

L(4)L(4) ≈\approx 2.32479−1.38452i2.32479 - 1.38452i
L(12)L(\frac12) ≈\approx 2.32479−1.38452i2.32479 - 1.38452i
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+(1.05e5+2.89e5i)T 1 + (1.05e5 + 2.89e5i)T
good3 1+(−27.4−47.5i)T+(−1.09e3+1.89e3i)T2 1 + (-27.4 - 47.5i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+(−8.21−14.2i)T+(−3.90e4+6.76e4i)T2 1 + (-8.21 - 14.2i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(392.+679.i)T+(−4.11e5+7.13e5i)T2 1 + (392. + 679. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1−6.02e3T+1.94e7T2 1 - 6.02e3T + 1.94e7T^{2}
13 1+(6.90e3+1.19e4i)T+(−3.13e7+5.43e7i)T2 1 + (6.90e3 + 1.19e4i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+(−6.83e3+1.18e4i)T+(−2.05e8−3.55e8i)T2 1 + (-6.83e3 + 1.18e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(−2.00e4−3.46e4i)T+(−4.46e8+7.74e8i)T2 1 + (-2.00e4 - 3.46e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1−7.04e4T+3.40e9T2 1 - 7.04e4T + 3.40e9T^{2}
29 1−1.94e5T+1.72e10T2 1 - 1.94e5T + 1.72e10T^{2}
31 1+2.34e5T+2.75e10T2 1 + 2.34e5T + 2.75e10T^{2}
41 1+(1.18e5+2.05e5i)T+(−9.73e10+1.68e11i)T2 1 + (1.18e5 + 2.05e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+8.33e5T+2.71e11T2 1 + 8.33e5T + 2.71e11T^{2}
47 1+1.45e5T+5.06e11T2 1 + 1.45e5T + 5.06e11T^{2}
53 1+(−5.42e5+9.40e5i)T+(−5.87e11−1.01e12i)T2 1 + (-5.42e5 + 9.40e5i)T + (-5.87e11 - 1.01e12i)T^{2}
59 1+(−1.74e5+3.02e5i)T+(−1.24e12−2.15e12i)T2 1 + (-1.74e5 + 3.02e5i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−1.01e6−1.76e6i)T+(−1.57e12+2.72e12i)T2 1 + (-1.01e6 - 1.76e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−1.08e6−1.88e6i)T+(−3.03e12+5.24e12i)T2 1 + (-1.08e6 - 1.88e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(−1.65e6−2.87e6i)T+(−4.54e12+7.87e12i)T2 1 + (-1.65e6 - 2.87e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1+6.03e6T+1.10e13T2 1 + 6.03e6T + 1.10e13T^{2}
79 1+(1.59e6+2.76e6i)T+(−9.60e12+1.66e13i)T2 1 + (1.59e6 + 2.76e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(4.32e6−7.48e6i)T+(−1.35e13−2.35e13i)T2 1 + (4.32e6 - 7.48e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+(−5.00e6+8.66e6i)T+(−2.21e13−3.83e13i)T2 1 + (-5.00e6 + 8.66e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+1.05e7T+8.07e13T2 1 + 1.05e7T + 8.07e13T^{2}
show more
show less
   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

−12.90607023130519995301399565422, −11.87264347584420175303701326718, −10.31612018141555910341550480890, −9.939138340100285604684442473839, −8.731073428067965931674959884640, −7.01307188014914454903071901276, −5.20250502720281223881223604658, −3.85793815915789452736238181697, −3.05056615658713228902102674716, −0.853656670543043251150904843106, 1.51965057226700115378126088352, 3.04582672799227104412149028817, 4.85133164697781927387077920642, 6.61281999392635242634230364253, 7.10768899868243309937872561407, 8.698116354476087081192345411931, 9.366144160792028117041069351232, 11.62238910873377996261121693283, 12.43613169729054438852588780212, 13.43283785243265705569541900226

Graph of the ZZ-function along the critical line