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Properties

Label 2-63-9.7-c7-0-32
Degree 22
Conductor 6363
Sign −0.132+0.991i-0.132 + 0.991i
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  + (8.53 − 14.7i)2-s + (46.7 − 0.978i)3-s + (−81.6 − 141. i)4-s + (117. + 202. i)5-s + (384. − 699. i)6-s + (−171.5 + 297. i)7-s − 601.·8-s + (2.18e3 − 91.4i)9-s + 3.99e3·10-s + (3.33e3 − 5.78e3i)11-s + (−3.95e3 − 6.53e3i)12-s + (−4.94e3 − 8.57e3i)13-s + (2.92e3 + 5.06e3i)14-s + (5.67e3 + 9.37e3i)15-s + (5.31e3 − 9.20e3i)16-s − 1.67e4·17-s + ⋯
L(s)  = 1  + (0.754 − 1.30i)2-s + (0.999 − 0.0209i)3-s + (−0.637 − 1.10i)4-s + (0.419 + 0.726i)5-s + (0.726 − 1.32i)6-s + (−0.188 + 0.327i)7-s − 0.415·8-s + (0.999 − 0.0418i)9-s + 1.26·10-s + (0.756 − 1.31i)11-s + (−0.660 − 1.09i)12-s + (−0.624 − 1.08i)13-s + (0.285 + 0.493i)14-s + (0.434 + 0.717i)15-s + (0.324 − 0.561i)16-s − 0.827·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.132+0.991i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((−0.132+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.132+0.991i-0.132 + 0.991i
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), −0.132+0.991i)(2,\ 63,\ (\ :7/2),\ -0.132 + 0.991i)

Particular Values

L(4)L(4) ≈\approx 2.87408−3.28321i2.87408 - 3.28321i
L(12)L(\frac12) ≈\approx 2.87408−3.28321i2.87408 - 3.28321i
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+(−46.7+0.978i)T 1 + (-46.7 + 0.978i)T
7 1+(171.5−297.i)T 1 + (171.5 - 297. i)T
good2 1+(−8.53+14.7i)T+(−64−110.i)T2 1 + (-8.53 + 14.7i)T + (-64 - 110. i)T^{2}
5 1+(−117.−202.i)T+(−3.90e4+6.76e4i)T2 1 + (-117. - 202. i)T + (-3.90e4 + 6.76e4i)T^{2}
11 1+(−3.33e3+5.78e3i)T+(−9.74e6−1.68e7i)T2 1 + (-3.33e3 + 5.78e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(4.94e3+8.57e3i)T+(−3.13e7+5.43e7i)T2 1 + (4.94e3 + 8.57e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+1.67e4T+4.10e8T2 1 + 1.67e4T + 4.10e8T^{2}
19 1−4.98e4T+8.93e8T2 1 - 4.98e4T + 8.93e8T^{2}
23 1+(−2.61e4−4.53e4i)T+(−1.70e9+2.94e9i)T2 1 + (-2.61e4 - 4.53e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(1.27e5−2.20e5i)T+(−8.62e9−1.49e10i)T2 1 + (1.27e5 - 2.20e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+(−7.56e4−1.31e5i)T+(−1.37e10+2.38e10i)T2 1 + (-7.56e4 - 1.31e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+5.66e5T+9.49e10T2 1 + 5.66e5T + 9.49e10T^{2}
41 1+(1.52e4+2.63e4i)T+(−9.73e10+1.68e11i)T2 1 + (1.52e4 + 2.63e4i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(2.82e5−4.89e5i)T+(−1.35e11−2.35e11i)T2 1 + (2.82e5 - 4.89e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+(−7.80e4+1.35e5i)T+(−2.53e11−4.38e11i)T2 1 + (-7.80e4 + 1.35e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1−3.30e5T+1.17e12T2 1 - 3.30e5T + 1.17e12T^{2}
59 1+(−2.06e5−3.58e5i)T+(−1.24e12+2.15e12i)T2 1 + (-2.06e5 - 3.58e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(5.89e5−1.02e6i)T+(−1.57e12−2.72e12i)T2 1 + (5.89e5 - 1.02e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(−5.80e5−1.00e6i)T+(−3.03e12+5.24e12i)T2 1 + (-5.80e5 - 1.00e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+3.64e6T+9.09e12T2 1 + 3.64e6T + 9.09e12T^{2}
73 1−3.90e5T+1.10e13T2 1 - 3.90e5T + 1.10e13T^{2}
79 1+(2.66e6−4.61e6i)T+(−9.60e12−1.66e13i)T2 1 + (2.66e6 - 4.61e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(−2.96e6+5.13e6i)T+(−1.35e13−2.35e13i)T2 1 + (-2.96e6 + 5.13e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1−9.78e4T+4.42e13T2 1 - 9.78e4T + 4.42e13T^{2}
97 1+(2.68e6−4.65e6i)T+(−4.03e13−6.99e13i)T2 1 + (2.68e6 - 4.65e6i)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

−13.31879191038108700306240864205, −12.16438477177205358851241356066, −10.97432712883316485833951810293, −9.992332527655170097066935349031, −8.852802180001710085472917557699, −7.13894290116044254110933761017, −5.30675937853786609486688429899, −3.37933169238087808242792152593, −2.92835581780801126289277623479, −1.36255357384240756881360490485, 1.81526123988989159044856730374, 4.05327040207265340480746625931, 4.93565990112440573594995267194, 6.74801193006197639717857714672, 7.48184479109248150799174488780, 8.985529379817370399691612125754, 9.789521848152598115764183050960, 12.11752313996752501076253029409, 13.27992436111741196923710191899, 13.90022049950359144201040475701

Graph of the ZZ-function along the critical line