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Properties

Label 2-63-9.7-c7-0-13
Degree 22
Conductor 6363
Sign 0.423−0.905i0.423 - 0.905i
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  + (−7.85 + 13.5i)2-s + (−6.13 + 46.3i)3-s + (−59.2 − 102. i)4-s + (−213. − 370. i)5-s + (−582. − 447. i)6-s + (−171.5 + 297. i)7-s − 148.·8-s + (−2.11e3 − 568. i)9-s + 6.71e3·10-s + (1.55e3 − 2.68e3i)11-s + (5.12e3 − 2.11e3i)12-s + (3.65e3 + 6.32e3i)13-s + (−2.69e3 − 4.66e3i)14-s + (1.84e4 − 7.64e3i)15-s + (8.75e3 − 1.51e4i)16-s + 2.24e4·17-s + ⋯
L(s)  = 1  + (−0.693 + 1.20i)2-s + (−0.131 + 0.991i)3-s + (−0.463 − 0.802i)4-s + (−0.765 − 1.32i)5-s + (−1.10 − 0.845i)6-s + (−0.188 + 0.327i)7-s − 0.102·8-s + (−0.965 − 0.259i)9-s + 2.12·10-s + (0.351 − 0.608i)11-s + (0.856 − 0.353i)12-s + (0.461 + 0.798i)13-s + (−0.262 − 0.454i)14-s + (1.41 − 0.584i)15-s + (0.534 − 0.925i)16-s + 1.10·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.423−0.905i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((0.423−0.905i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.423−0.905i0.423 - 0.905i
Analytic conductor: 19.680219.6802
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ63(43,⋅)\chi_{63} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :7/2), 0.423−0.905i)(2,\ 63,\ (\ :7/2),\ 0.423 - 0.905i)

Particular Values

L(4)L(4) ≈\approx 0.641114+0.407889i0.641114 + 0.407889i
L(12)L(\frac12) ≈\approx 0.641114+0.407889i0.641114 + 0.407889i
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+(6.13−46.3i)T 1 + (6.13 - 46.3i)T
7 1+(171.5−297.i)T 1 + (171.5 - 297. i)T
good2 1+(7.85−13.5i)T+(−64−110.i)T2 1 + (7.85 - 13.5i)T + (-64 - 110. i)T^{2}
5 1+(213.+370.i)T+(−3.90e4+6.76e4i)T2 1 + (213. + 370. i)T + (-3.90e4 + 6.76e4i)T^{2}
11 1+(−1.55e3+2.68e3i)T+(−9.74e6−1.68e7i)T2 1 + (-1.55e3 + 2.68e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(−3.65e3−6.32e3i)T+(−3.13e7+5.43e7i)T2 1 + (-3.65e3 - 6.32e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1−2.24e4T+4.10e8T2 1 - 2.24e4T + 4.10e8T^{2}
19 1+4.75e4T+8.93e8T2 1 + 4.75e4T + 8.93e8T^{2}
23 1+(−2.14e4−3.71e4i)T+(−1.70e9+2.94e9i)T2 1 + (-2.14e4 - 3.71e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−7.41e4+1.28e5i)T+(−8.62e9−1.49e10i)T2 1 + (-7.41e4 + 1.28e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+(−3.59e4−6.22e4i)T+(−1.37e10+2.38e10i)T2 1 + (-3.59e4 - 6.22e4i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+2.48e5T+9.49e10T2 1 + 2.48e5T + 9.49e10T^{2}
41 1+(−1.90e5−3.29e5i)T+(−9.73e10+1.68e11i)T2 1 + (-1.90e5 - 3.29e5i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(3.22e4−5.59e4i)T+(−1.35e11−2.35e11i)T2 1 + (3.22e4 - 5.59e4i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+(−3.86e5+6.68e5i)T+(−2.53e11−4.38e11i)T2 1 + (-3.86e5 + 6.68e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1−6.93e5T+1.17e12T2 1 - 6.93e5T + 1.17e12T^{2}
59 1+(1.14e6+1.99e6i)T+(−1.24e12+2.15e12i)T2 1 + (1.14e6 + 1.99e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(−1.54e6+2.67e6i)T+(−1.57e12−2.72e12i)T2 1 + (-1.54e6 + 2.67e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(−1.36e6−2.35e6i)T+(−3.03e12+5.24e12i)T2 1 + (-1.36e6 - 2.35e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1−3.24e6T+9.09e12T2 1 - 3.24e6T + 9.09e12T^{2}
73 1−5.42e6T+1.10e13T2 1 - 5.42e6T + 1.10e13T^{2}
79 1+(7.25e5−1.25e6i)T+(−9.60e12−1.66e13i)T2 1 + (7.25e5 - 1.25e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(2.46e6−4.26e6i)T+(−1.35e13−2.35e13i)T2 1 + (2.46e6 - 4.26e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1−5.52e5T+4.42e13T2 1 - 5.52e5T + 4.42e13T^{2}
97 1+(−7.76e6+1.34e7i)T+(−4.03e13−6.99e13i)T2 1 + (-7.76e6 + 1.34e7i)T + (-4.03e13 - 6.99e13i)T^{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

−14.17940397535432725663875754060, −12.41518701947418602156452822428, −11.42241597544030567123126657915, −9.698341861012721954357240099294, −8.721264227433734350372468261148, −8.226748408399471514994570899158, −6.35698095685786821724674642136, −5.15667453747335591999756538708, −3.78149014840310515361297115428, −0.53282401839442933872041393357, 0.829142372437698776376442125492, 2.42423941331436471406450847910, 3.53341653079962753662446002230, 6.30419446794609144971303479569, 7.41677535127220358480670322517, 8.599702802088363989684613481132, 10.39733160197207376306932462389, 10.86948614780426279804254799871, 12.04507330295961371814585592729, 12.74698334928923098933639563929

Graph of the ZZ-function along the critical line