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Properties

Label 2-63-7.2-c9-0-14
Degree 22
Conductor 6363
Sign 0.386+0.922i0.386 + 0.922i
Analytic cond. 32.447232.4472
Root an. cond. 5.696245.69624
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.4 − 33.7i)2-s + (−502. − 871. i)4-s + (−1.23e3 + 2.14e3i)5-s + (6.00e3 − 2.08e3i)7-s − 1.92e4·8-s + (4.82e4 + 8.35e4i)10-s + (9.74e3 + 1.68e4i)11-s + 2.30e4·13-s + (4.67e4 − 2.43e5i)14-s + (−1.17e5 + 2.03e5i)16-s + (2.82e5 + 4.88e5i)17-s + (2.59e5 − 4.49e5i)19-s + 2.49e6·20-s + 7.59e5·22-s + (7.72e5 − 1.33e6i)23-s + ⋯
L(s)  = 1  + (0.860 − 1.49i)2-s + (−0.982 − 1.70i)4-s + (−0.886 + 1.53i)5-s + (0.944 − 0.327i)7-s − 1.66·8-s + (1.52 + 2.64i)10-s + (0.200 + 0.347i)11-s + 0.224·13-s + (0.325 − 1.69i)14-s + (−0.447 + 0.775i)16-s + (0.818 + 1.41i)17-s + (0.457 − 0.791i)19-s + 3.48·20-s + 0.690·22-s + (0.575 − 0.997i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.386+0.922i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+9/2) L(s)=((0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.386+0.922i0.386 + 0.922i
Analytic conductor: 32.447232.4472
Root analytic conductor: 5.696245.69624
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ63(37,⋅)\chi_{63} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :9/2), 0.386+0.922i)(2,\ 63,\ (\ :9/2),\ 0.386 + 0.922i)

Particular Values

L(5)L(5) ≈\approx 2.50162−1.66379i2.50162 - 1.66379i
L(12)L(\frac12) ≈\approx 2.50162−1.66379i2.50162 - 1.66379i
L(112)L(\frac{11}{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+(−6.00e3+2.08e3i)T 1 + (-6.00e3 + 2.08e3i)T
good2 1+(−19.4+33.7i)T+(−256−443.i)T2 1 + (-19.4 + 33.7i)T + (-256 - 443. i)T^{2}
5 1+(1.23e3−2.14e3i)T+(−9.76e5−1.69e6i)T2 1 + (1.23e3 - 2.14e3i)T + (-9.76e5 - 1.69e6i)T^{2}
11 1+(−9.74e3−1.68e4i)T+(−1.17e9+2.04e9i)T2 1 + (-9.74e3 - 1.68e4i)T + (-1.17e9 + 2.04e9i)T^{2}
13 1−2.30e4T+1.06e10T2 1 - 2.30e4T + 1.06e10T^{2}
17 1+(−2.82e5−4.88e5i)T+(−5.92e10+1.02e11i)T2 1 + (-2.82e5 - 4.88e5i)T + (-5.92e10 + 1.02e11i)T^{2}
19 1+(−2.59e5+4.49e5i)T+(−1.61e11−2.79e11i)T2 1 + (-2.59e5 + 4.49e5i)T + (-1.61e11 - 2.79e11i)T^{2}
23 1+(−7.72e5+1.33e6i)T+(−9.00e11−1.55e12i)T2 1 + (-7.72e5 + 1.33e6i)T + (-9.00e11 - 1.55e12i)T^{2}
29 1−4.76e6T+1.45e13T2 1 - 4.76e6T + 1.45e13T^{2}
31 1+(−2.35e6−4.08e6i)T+(−1.32e13+2.28e13i)T2 1 + (-2.35e6 - 4.08e6i)T + (-1.32e13 + 2.28e13i)T^{2}
37 1+(9.94e6−1.72e7i)T+(−6.49e13−1.12e14i)T2 1 + (9.94e6 - 1.72e7i)T + (-6.49e13 - 1.12e14i)T^{2}
41 1−1.93e7T+3.27e14T2 1 - 1.93e7T + 3.27e14T^{2}
43 1−9.57e5T+5.02e14T2 1 - 9.57e5T + 5.02e14T^{2}
47 1+(1.71e7−2.97e7i)T+(−5.59e14−9.69e14i)T2 1 + (1.71e7 - 2.97e7i)T + (-5.59e14 - 9.69e14i)T^{2}
53 1+(−1.01e7−1.75e7i)T+(−1.64e15+2.85e15i)T2 1 + (-1.01e7 - 1.75e7i)T + (-1.64e15 + 2.85e15i)T^{2}
59 1+(6.21e6+1.07e7i)T+(−4.33e15+7.50e15i)T2 1 + (6.21e6 + 1.07e7i)T + (-4.33e15 + 7.50e15i)T^{2}
61 1+(−8.00e7+1.38e8i)T+(−5.84e15−1.01e16i)T2 1 + (-8.00e7 + 1.38e8i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(−3.39e7−5.88e7i)T+(−1.36e16+2.35e16i)T2 1 + (-3.39e7 - 5.88e7i)T + (-1.36e16 + 2.35e16i)T^{2}
71 1−2.84e8T+4.58e16T2 1 - 2.84e8T + 4.58e16T^{2}
73 1+(−1.30e8−2.26e8i)T+(−2.94e16+5.09e16i)T2 1 + (-1.30e8 - 2.26e8i)T + (-2.94e16 + 5.09e16i)T^{2}
79 1+(4.18e7−7.25e7i)T+(−5.99e16−1.03e17i)T2 1 + (4.18e7 - 7.25e7i)T + (-5.99e16 - 1.03e17i)T^{2}
83 1+1.69e7T+1.86e17T2 1 + 1.69e7T + 1.86e17T^{2}
89 1+(−5.07e7+8.78e7i)T+(−1.75e17−3.03e17i)T2 1 + (-5.07e7 + 8.78e7i)T + (-1.75e17 - 3.03e17i)T^{2}
97 1−8.99e8T+7.60e17T2 1 - 8.99e8T + 7.60e17T^{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

−12.53057445015252983121739446682, −11.60233135626335828723701267336, −10.82898641070659082680249477441, −10.20876934652773416366927665335, −8.127561600084179093342251201393, −6.66190219131647015316276787398, −4.76093235905677142088362011175, −3.67087886334730060646385634896, −2.65206715669405073593106356705, −1.12800689203215577597028500405, 0.914910810276086381483152321107, 3.77636222587912474958590271424, 4.92715911574459564162440570211, 5.55685848509580674421774426140, 7.44956207370274259754484330149, 8.157464275081787314992244236692, 9.115309302675675713787944072645, 11.64210304208516003529320861155, 12.32982951215497133249809720584, 13.54678716604371559666046169731

Graph of the ZZ-function along the critical line