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Properties

Label 2-63-7.2-c7-0-6
Degree 22
Conductor 6363
Sign 0.106−0.994i0.106 - 0.994i
Analytic cond. 19.680219.6802
Root an. cond. 4.436244.43624
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  + (3.40 − 5.89i)2-s + (40.8 + 70.7i)4-s + (6.28 − 10.8i)5-s + (−894. − 153. i)7-s + 1.42e3·8-s + (−42.7 − 73.9i)10-s + (3.35e3 + 5.81e3i)11-s − 8.77e3·13-s + (−3.94e3 + 4.74e3i)14-s + (−378. + 655. i)16-s + (−6.96e3 − 1.20e4i)17-s + (−1.71e4 + 2.97e4i)19-s + 1.02e3·20-s + 4.57e4·22-s + (−4.22e4 + 7.31e4i)23-s + ⋯
L(s)  = 1  + (0.300 − 0.520i)2-s + (0.319 + 0.552i)4-s + (0.0224 − 0.0389i)5-s + (−0.985 − 0.169i)7-s + 0.985·8-s + (−0.0135 − 0.0233i)10-s + (0.761 + 1.31i)11-s − 1.10·13-s + (−0.384 + 0.462i)14-s + (−0.0231 + 0.0400i)16-s + (−0.343 − 0.595i)17-s + (−0.575 + 0.996i)19-s + 0.0286·20-s + 0.915·22-s + (−0.724 + 1.25i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.106−0.994i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((0.106−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.106−0.994i0.106 - 0.994i
Analytic conductor: 19.680219.6802
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ63(37,⋅)\chi_{63} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :7/2), 0.106−0.994i)(2,\ 63,\ (\ :7/2),\ 0.106 - 0.994i)

Particular Values

L(4)L(4) ≈\approx 1.21654+1.09276i1.21654 + 1.09276i
L(12)L(\frac12) ≈\approx 1.21654+1.09276i1.21654 + 1.09276i
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)
bad3 1 1
7 1+(894.+153.i)T 1 + (894. + 153. i)T
good2 1+(−3.40+5.89i)T+(−64−110.i)T2 1 + (-3.40 + 5.89i)T + (-64 - 110. i)T^{2}
5 1+(−6.28+10.8i)T+(−3.90e4−6.76e4i)T2 1 + (-6.28 + 10.8i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−3.35e3−5.81e3i)T+(−9.74e6+1.68e7i)T2 1 + (-3.35e3 - 5.81e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+8.77e3T+6.27e7T2 1 + 8.77e3T + 6.27e7T^{2}
17 1+(6.96e3+1.20e4i)T+(−2.05e8+3.55e8i)T2 1 + (6.96e3 + 1.20e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(1.71e4−2.97e4i)T+(−4.46e8−7.74e8i)T2 1 + (1.71e4 - 2.97e4i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+(4.22e4−7.31e4i)T+(−1.70e9−2.94e9i)T2 1 + (4.22e4 - 7.31e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1−1.09e5T+1.72e10T2 1 - 1.09e5T + 1.72e10T^{2}
31 1+(−7.70e4−1.33e5i)T+(−1.37e10+2.38e10i)T2 1 + (-7.70e4 - 1.33e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+(−7.65e4+1.32e5i)T+(−4.74e10−8.22e10i)T2 1 + (-7.65e4 + 1.32e5i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1−1.61e5T+1.94e11T2 1 - 1.61e5T + 1.94e11T^{2}
43 1+6.33e5T+2.71e11T2 1 + 6.33e5T + 2.71e11T^{2}
47 1+(7.83e4−1.35e5i)T+(−2.53e11−4.38e11i)T2 1 + (7.83e4 - 1.35e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(5.37e5+9.30e5i)T+(−5.87e11+1.01e12i)T2 1 + (5.37e5 + 9.30e5i)T + (-5.87e11 + 1.01e12i)T^{2}
59 1+(5.57e5+9.65e5i)T+(−1.24e12+2.15e12i)T2 1 + (5.57e5 + 9.65e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.12e6−1.94e6i)T+(−1.57e12−2.72e12i)T2 1 + (1.12e6 - 1.94e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(3.67e5+6.36e5i)T+(−3.03e12+5.24e12i)T2 1 + (3.67e5 + 6.36e5i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1−1.20e6T+9.09e12T2 1 - 1.20e6T + 9.09e12T^{2}
73 1+(3.09e6+5.36e6i)T+(−5.52e12+9.56e12i)T2 1 + (3.09e6 + 5.36e6i)T + (-5.52e12 + 9.56e12i)T^{2}
79 1+(−2.45e6+4.24e6i)T+(−9.60e12−1.66e13i)T2 1 + (-2.45e6 + 4.24e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1−7.43e6T+2.71e13T2 1 - 7.43e6T + 2.71e13T^{2}
89 1+(−5.93e5+1.02e6i)T+(−2.21e13−3.83e13i)T2 1 + (-5.93e5 + 1.02e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+3.93e6T+8.07e13T2 1 + 3.93e6T + 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

−13.47283469130527016071620668541, −12.41879124252332925716292165714, −11.90087889071431291616210348502, −10.32344423365499819384708419442, −9.406054270587112238421193453815, −7.59110156098108779411599724249, −6.66512075284171695965320220094, −4.61714282873216234169376470025, −3.32015849081514474601455111043, −1.86369500206933907292338679609, 0.51056083655295912217857597277, 2.59319370732128406579054309435, 4.48779838242372634312711509977, 6.13701590938347266837474711303, 6.69172719293595256822482720422, 8.457337284391272871124394697214, 9.830658241524903542950941484163, 10.87668394237478192188445768272, 12.21341797093785914751284682977, 13.48177236984458985376530199486

Graph of the ZZ-function along the critical line