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Properties

Label 2-74-37.10-c7-0-8
Degree 22
Conductor 7474
Sign 0.519+0.854i0.519 + 0.854i
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 + (−27.0 − 46.8i)3-s + (−31.9 − 55.4i)4-s + (61.0 + 105. i)5-s − 433.·6-s + (612. + 1.06e3i)7-s − 511.·8-s + (−371. + 643. i)9-s + 976.·10-s + 8.41e3·11-s + (−1.73e3 + 3.00e3i)12-s + (2.86e3 + 4.95e3i)13-s + 9.79e3·14-s + (3.30e3 − 5.72e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−3.81e3 + 6.60e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.578 − 1.00i)3-s + (−0.249 − 0.433i)4-s + (0.218 + 0.378i)5-s − 0.818·6-s + (0.674 + 1.16i)7-s − 0.353·8-s + (−0.169 + 0.294i)9-s + 0.308·10-s + 1.90·11-s + (−0.289 + 0.501i)12-s + (0.361 + 0.625i)13-s + 0.953·14-s + (0.252 − 0.437i)15-s + (−0.125 + 0.216i)16-s + (−0.188 + 0.325i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.519+0.854i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((0.519+0.854i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.519+0.854i0.519 + 0.854i
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), 0.519+0.854i)(2,\ 74,\ (\ :7/2),\ 0.519 + 0.854i)

Particular Values

L(4)L(4) ≈\approx 2.01426−1.13245i2.01426 - 1.13245i
L(12)L(\frac12) ≈\approx 2.01426−1.13245i2.01426 - 1.13245i
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+(1.19e5+2.83e5i)T 1 + (1.19e5 + 2.83e5i)T
good3 1+(27.0+46.8i)T+(−1.09e3+1.89e3i)T2 1 + (27.0 + 46.8i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+(−61.0−105.i)T+(−3.90e4+6.76e4i)T2 1 + (-61.0 - 105. i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(−612.−1.06e3i)T+(−4.11e5+7.13e5i)T2 1 + (-612. - 1.06e3i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1−8.41e3T+1.94e7T2 1 - 8.41e3T + 1.94e7T^{2}
13 1+(−2.86e3−4.95e3i)T+(−3.13e7+5.43e7i)T2 1 + (-2.86e3 - 4.95e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+(3.81e3−6.60e3i)T+(−2.05e8−3.55e8i)T2 1 + (3.81e3 - 6.60e3i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(−6.66e3−1.15e4i)T+(−4.46e8+7.74e8i)T2 1 + (-6.66e3 - 1.15e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+9.29e4T+3.40e9T2 1 + 9.29e4T + 3.40e9T^{2}
29 1−1.94e5T+1.72e10T2 1 - 1.94e5T + 1.72e10T^{2}
31 1−4.01e4T+2.75e10T2 1 - 4.01e4T + 2.75e10T^{2}
41 1+(−4.26e5−7.37e5i)T+(−9.73e10+1.68e11i)T2 1 + (-4.26e5 - 7.37e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1−1.00e6T+2.71e11T2 1 - 1.00e6T + 2.71e11T^{2}
47 1+6.11e5T+5.06e11T2 1 + 6.11e5T + 5.06e11T^{2}
53 1+(−4.15e5+7.20e5i)T+(−5.87e11−1.01e12i)T2 1 + (-4.15e5 + 7.20e5i)T + (-5.87e11 - 1.01e12i)T^{2}
59 1+(−1.26e6+2.19e6i)T+(−1.24e12−2.15e12i)T2 1 + (-1.26e6 + 2.19e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(1.31e5+2.28e5i)T+(−1.57e12+2.72e12i)T2 1 + (1.31e5 + 2.28e5i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−8.53e5−1.47e6i)T+(−3.03e12+5.24e12i)T2 1 + (-8.53e5 - 1.47e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(−4.56e5−7.91e5i)T+(−4.54e12+7.87e12i)T2 1 + (-4.56e5 - 7.91e5i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1+1.98e6T+1.10e13T2 1 + 1.98e6T + 1.10e13T^{2}
79 1+(1.41e6+2.44e6i)T+(−9.60e12+1.66e13i)T2 1 + (1.41e6 + 2.44e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(1.15e6−2.00e6i)T+(−1.35e13−2.35e13i)T2 1 + (1.15e6 - 2.00e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+(4.52e6−7.84e6i)T+(−2.21e13−3.83e13i)T2 1 + (4.52e6 - 7.84e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+1.55e7T+8.07e13T2 1 + 1.55e7T + 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

−12.52376871716852797659896373097, −11.92577903316085488389917954025, −11.29144129499507629743939362086, −9.627545802454106466562133026449, −8.413878564851907714225329539247, −6.56973967743199412380590551454, −5.95905142937234605423555197276, −4.14800057202212877224135851619, −2.16059720699662407091108010526, −1.18162310942227803836733542089, 1.00508528616452608598175543655, 3.92498371040600341897429673864, 4.59904749643555784916069684511, 5.93292669483692639491905300907, 7.29316910116129920005422174409, 8.768035809238113754456304402905, 9.979696272647546219108381878250, 11.06336580281585156361648487620, 12.11386264445556975146959587786, 13.70474880114004099936355195136

Graph of the ZZ-function along the critical line