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Properties

Label 2-74-37.10-c7-0-17
Degree 22
Conductor 7474
Sign −0.864+0.502i-0.864 + 0.502i
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 + (34.1 + 59.2i)3-s + (−31.9 − 55.4i)4-s + (13.0 + 22.6i)5-s + 546.·6-s + (−837. − 1.45e3i)7-s − 511.·8-s + (−1.24e3 + 2.15e3i)9-s + 208.·10-s − 5.50e3·11-s + (2.18e3 − 3.78e3i)12-s + (3.58e3 + 6.21e3i)13-s − 1.34e4·14-s + (−892. + 1.54e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (509. − 883. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.730 + 1.26i)3-s + (−0.249 − 0.433i)4-s + (0.0466 + 0.0808i)5-s + 1.03·6-s + (−0.923 − 1.59i)7-s − 0.353·8-s + (−0.568 + 0.985i)9-s + 0.0660·10-s − 1.24·11-s + (0.365 − 0.633i)12-s + (0.453 + 0.784i)13-s − 1.30·14-s + (−0.0682 + 0.118i)15-s + (−0.125 + 0.216i)16-s + (0.0251 − 0.0435i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.864+0.502i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.864+0.502i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.864+0.502i-0.864 + 0.502i
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), −0.864+0.502i)(2,\ 74,\ (\ :7/2),\ -0.864 + 0.502i)

Particular Values

L(4)L(4) ≈\approx 0.229878−0.852599i0.229878 - 0.852599i
L(12)L(\frac12) ≈\approx 0.229878−0.852599i0.229878 - 0.852599i
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+(−2.86e5+1.14e5i)T 1 + (-2.86e5 + 1.14e5i)T
good3 1+(−34.1−59.2i)T+(−1.09e3+1.89e3i)T2 1 + (-34.1 - 59.2i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+(−13.0−22.6i)T+(−3.90e4+6.76e4i)T2 1 + (-13.0 - 22.6i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(837.+1.45e3i)T+(−4.11e5+7.13e5i)T2 1 + (837. + 1.45e3i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+5.50e3T+1.94e7T2 1 + 5.50e3T + 1.94e7T^{2}
13 1+(−3.58e3−6.21e3i)T+(−3.13e7+5.43e7i)T2 1 + (-3.58e3 - 6.21e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+(−509.+883.i)T+(−2.05e8−3.55e8i)T2 1 + (-509. + 883. i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(2.69e4+4.67e4i)T+(−4.46e8+7.74e8i)T2 1 + (2.69e4 + 4.67e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+1.06e5T+3.40e9T2 1 + 1.06e5T + 3.40e9T^{2}
29 1+6.96e4T+1.72e10T2 1 + 6.96e4T + 1.72e10T^{2}
31 1−1.61e5T+2.75e10T2 1 - 1.61e5T + 2.75e10T^{2}
41 1+(−3.50e4−6.06e4i)T+(−9.73e10+1.68e11i)T2 1 + (-3.50e4 - 6.06e4i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+5.96e5T+2.71e11T2 1 + 5.96e5T + 2.71e11T^{2}
47 1+1.09e6T+5.06e11T2 1 + 1.09e6T + 5.06e11T^{2}
53 1+(−1.79e5+3.10e5i)T+(−5.87e11−1.01e12i)T2 1 + (-1.79e5 + 3.10e5i)T + (-5.87e11 - 1.01e12i)T^{2}
59 1+(−1.38e6+2.39e6i)T+(−1.24e12−2.15e12i)T2 1 + (-1.38e6 + 2.39e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−1.38e6−2.40e6i)T+(−1.57e12+2.72e12i)T2 1 + (-1.38e6 - 2.40e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−1.30e6−2.25e6i)T+(−3.03e12+5.24e12i)T2 1 + (-1.30e6 - 2.25e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(1.63e5+2.83e5i)T+(−4.54e12+7.87e12i)T2 1 + (1.63e5 + 2.83e5i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1+2.90e6T+1.10e13T2 1 + 2.90e6T + 1.10e13T^{2}
79 1+(−3.32e5−5.76e5i)T+(−9.60e12+1.66e13i)T2 1 + (-3.32e5 - 5.76e5i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(−1.29e6+2.24e6i)T+(−1.35e13−2.35e13i)T2 1 + (-1.29e6 + 2.24e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+(2.19e6−3.79e6i)T+(−2.21e13−3.83e13i)T2 1 + (2.19e6 - 3.79e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1−1.04e6T+8.07e13T2 1 - 1.04e6T + 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

−13.05885652368321602763072045436, −11.17940212893923547643854832067, −10.23547655600936257405001763470, −9.774447699959732133396469876034, −8.339638234860288968544222036435, −6.61681056783653010291888318701, −4.60105894034061668956106717707, −3.85322733940656867632648441464, −2.62905330795704709210735565687, −0.22376852600119187155226881562, 2.10316931988818556844018319946, 3.20062714827599164440517657547, 5.61408462412428829432984451682, 6.38410084061066073926577466461, 8.002045301139197254474714184916, 8.401782421455513318649939752053, 9.944431478748567948985240383450, 12.05142190235471962304991223187, 12.83958039609908379127755367542, 13.29515245160998271824835178101

Graph of the ZZ-function along the critical line