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Properties

Label 2-74-37.10-c7-0-15
Degree 22
Conductor 7474
Sign −0.926+0.377i-0.926 + 0.377i
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 + (−13.7 − 23.8i)3-s + (−31.9 − 55.4i)4-s + (−121. − 210. i)5-s + 220.·6-s + (208. + 360. i)7-s + 511.·8-s + (715. − 1.23e3i)9-s + 1.94e3·10-s + 4.33e3·11-s + (−880. + 1.52e3i)12-s + (−3.13e3 − 5.43e3i)13-s − 3.32e3·14-s + (−3.33e3 + 5.78e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−2.31e3 + 4.01e3i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.294 − 0.509i)3-s + (−0.249 − 0.433i)4-s + (−0.434 − 0.752i)5-s + 0.415·6-s + (0.229 + 0.397i)7-s + 0.353·8-s + (0.327 − 0.566i)9-s + 0.614·10-s + 0.983·11-s + (−0.147 + 0.254i)12-s + (−0.395 − 0.685i)13-s − 0.324·14-s + (−0.255 + 0.442i)15-s + (−0.125 + 0.216i)16-s + (−0.114 + 0.198i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((−0.926+0.377i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+7/2) L(s)=((−0.926+0.377i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: −0.926+0.377i-0.926 + 0.377i
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), −0.926+0.377i)(2,\ 74,\ (\ :7/2),\ -0.926 + 0.377i)

Particular Values

L(4)L(4) ≈\approx 0.0918907−0.469389i0.0918907 - 0.469389i
L(12)L(\frac12) ≈\approx 0.0918907−0.469389i0.0918907 - 0.469389i
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+(2.99e5−7.31e4i)T 1 + (2.99e5 - 7.31e4i)T
good3 1+(13.7+23.8i)T+(−1.09e3+1.89e3i)T2 1 + (13.7 + 23.8i)T + (-1.09e3 + 1.89e3i)T^{2}
5 1+(121.+210.i)T+(−3.90e4+6.76e4i)T2 1 + (121. + 210. i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(−208.−360.i)T+(−4.11e5+7.13e5i)T2 1 + (-208. - 360. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1−4.33e3T+1.94e7T2 1 - 4.33e3T + 1.94e7T^{2}
13 1+(3.13e3+5.43e3i)T+(−3.13e7+5.43e7i)T2 1 + (3.13e3 + 5.43e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+(2.31e3−4.01e3i)T+(−2.05e8−3.55e8i)T2 1 + (2.31e3 - 4.01e3i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(−5.77e3−1.00e4i)T+(−4.46e8+7.74e8i)T2 1 + (-5.77e3 - 1.00e4i)T + (-4.46e8 + 7.74e8i)T^{2}
23 1+1.08e5T+3.40e9T2 1 + 1.08e5T + 3.40e9T^{2}
29 1−1.09e5T+1.72e10T2 1 - 1.09e5T + 1.72e10T^{2}
31 1+1.25e5T+2.75e10T2 1 + 1.25e5T + 2.75e10T^{2}
41 1+(2.83e5+4.91e5i)T+(−9.73e10+1.68e11i)T2 1 + (2.83e5 + 4.91e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+4.19e5T+2.71e11T2 1 + 4.19e5T + 2.71e11T^{2}
47 1−1.90e4T+5.06e11T2 1 - 1.90e4T + 5.06e11T^{2}
53 1+(4.24e5−7.35e5i)T+(−5.87e11−1.01e12i)T2 1 + (4.24e5 - 7.35e5i)T + (-5.87e11 - 1.01e12i)T^{2}
59 1+(1.49e6−2.58e6i)T+(−1.24e12−2.15e12i)T2 1 + (1.49e6 - 2.58e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−1.49e5−2.58e5i)T+(−1.57e12+2.72e12i)T2 1 + (-1.49e5 - 2.58e5i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(4.21e5+7.29e5i)T+(−3.03e12+5.24e12i)T2 1 + (4.21e5 + 7.29e5i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+(−9.93e5−1.72e6i)T+(−4.54e12+7.87e12i)T2 1 + (-9.93e5 - 1.72e6i)T + (-4.54e12 + 7.87e12i)T^{2}
73 1+1.97e6T+1.10e13T2 1 + 1.97e6T + 1.10e13T^{2}
79 1+(−3.92e6−6.79e6i)T+(−9.60e12+1.66e13i)T2 1 + (-3.92e6 - 6.79e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(−7.76e5+1.34e6i)T+(−1.35e13−2.35e13i)T2 1 + (-7.76e5 + 1.34e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+(−1.28e6+2.22e6i)T+(−2.21e13−3.83e13i)T2 1 + (-1.28e6 + 2.22e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1−6.18e6T+8.07e13T2 1 - 6.18e6T + 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

−12.33399219982534323733263523611, −11.98310505485932239545782716263, −10.18450862679698061454208039797, −8.951729656748321263020902900231, −7.990738530653003916142114102146, −6.73363331917225936855429062474, −5.56700556866558605427578971815, −4.05524484062806988367497785786, −1.50389217266244190050390878927, −0.19870318377841487995455244270, 1.80349279069331571907097187564, 3.60202122850075881896939314543, 4.69075583361971147633901665449, 6.72393908685735298408586584090, 7.87608447420464397679000527137, 9.410673124927170393699457326219, 10.38794496829533104637433400315, 11.28673059866810894537941045978, 12.06995292009200113678006369471, 13.67572867705205614826767996495

Graph of the ZZ-function along the critical line