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Properties

Label 2-63-9.7-c7-0-1
Degree 22
Conductor 6363
Sign −0.366+0.930i-0.366 + 0.930i
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  + (−4.18 + 7.24i)2-s + (−46.5 − 4.69i)3-s + (28.9 + 50.1i)4-s + (47.2 + 81.8i)5-s + (228. − 317. i)6-s + (−171.5 + 297. i)7-s − 1.55e3·8-s + (2.14e3 + 436. i)9-s − 790.·10-s + (−3.76e3 + 6.51e3i)11-s + (−1.11e3 − 2.47e3i)12-s + (5.72e3 + 9.91e3i)13-s + (−1.43e3 − 2.48e3i)14-s + (−1.81e3 − 4.02e3i)15-s + (2.80e3 − 4.85e3i)16-s + 1.59e4·17-s + ⋯
L(s)  = 1  + (−0.369 + 0.640i)2-s + (−0.994 − 0.100i)3-s + (0.226 + 0.392i)4-s + (0.169 + 0.292i)5-s + (0.432 − 0.600i)6-s + (−0.188 + 0.327i)7-s − 1.07·8-s + (0.979 + 0.199i)9-s − 0.250·10-s + (−0.852 + 1.47i)11-s + (−0.185 − 0.412i)12-s + (0.722 + 1.25i)13-s + (−0.139 − 0.242i)14-s + (−0.138 − 0.308i)15-s + (0.171 − 0.296i)16-s + 0.789·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.366+0.930i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((−0.366+0.930i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.366+0.930i-0.366 + 0.930i
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), −0.366+0.930i)(2,\ 63,\ (\ :7/2),\ -0.366 + 0.930i)

Particular Values

L(4)L(4) ≈\approx 0.222800−0.327336i0.222800 - 0.327336i
L(12)L(\frac12) ≈\approx 0.222800−0.327336i0.222800 - 0.327336i
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+(46.5+4.69i)T 1 + (46.5 + 4.69i)T
7 1+(171.5−297.i)T 1 + (171.5 - 297. i)T
good2 1+(4.18−7.24i)T+(−64−110.i)T2 1 + (4.18 - 7.24i)T + (-64 - 110. i)T^{2}
5 1+(−47.2−81.8i)T+(−3.90e4+6.76e4i)T2 1 + (-47.2 - 81.8i)T + (-3.90e4 + 6.76e4i)T^{2}
11 1+(3.76e3−6.51e3i)T+(−9.74e6−1.68e7i)T2 1 + (3.76e3 - 6.51e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(−5.72e3−9.91e3i)T+(−3.13e7+5.43e7i)T2 1 + (-5.72e3 - 9.91e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1−1.59e4T+4.10e8T2 1 - 1.59e4T + 4.10e8T^{2}
19 1+3.59e4T+8.93e8T2 1 + 3.59e4T + 8.93e8T^{2}
23 1+(4.27e4+7.40e4i)T+(−1.70e9+2.94e9i)T2 1 + (4.27e4 + 7.40e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(2.03e3−3.52e3i)T+(−8.62e9−1.49e10i)T2 1 + (2.03e3 - 3.52e3i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+(−9.61e3−1.66e4i)T+(−1.37e10+2.38e10i)T2 1 + (-9.61e3 - 1.66e4i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1−2.78e5T+9.49e10T2 1 - 2.78e5T + 9.49e10T^{2}
41 1+(5.60e4+9.71e4i)T+(−9.73e10+1.68e11i)T2 1 + (5.60e4 + 9.71e4i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(−1.27e5+2.21e5i)T+(−1.35e11−2.35e11i)T2 1 + (-1.27e5 + 2.21e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+(2.86e5−4.95e5i)T+(−2.53e11−4.38e11i)T2 1 + (2.86e5 - 4.95e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+1.44e6T+1.17e12T2 1 + 1.44e6T + 1.17e12T^{2}
59 1+(1.37e6+2.38e6i)T+(−1.24e12+2.15e12i)T2 1 + (1.37e6 + 2.38e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(6.41e5−1.11e6i)T+(−1.57e12−2.72e12i)T2 1 + (6.41e5 - 1.11e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(−1.97e6−3.42e6i)T+(−3.03e12+5.24e12i)T2 1 + (-1.97e6 - 3.42e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1−1.16e6T+9.09e12T2 1 - 1.16e6T + 9.09e12T^{2}
73 1−1.35e6T+1.10e13T2 1 - 1.35e6T + 1.10e13T^{2}
79 1+(−3.36e6+5.82e6i)T+(−9.60e12−1.66e13i)T2 1 + (-3.36e6 + 5.82e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(3.24e6−5.61e6i)T+(−1.35e13−2.35e13i)T2 1 + (3.24e6 - 5.61e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1+8.78e6T+4.42e13T2 1 + 8.78e6T + 4.42e13T^{2}
97 1+(−3.21e6+5.56e6i)T+(−4.03e13−6.99e13i)T2 1 + (-3.21e6 + 5.56e6i)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

−14.55091142305569291353607473123, −12.74111000157605755337996769729, −12.19147961494053552683045728436, −10.86543244104647200079751716697, −9.693635522715906406100492686624, −8.150598209545749346900239316519, −6.87023684262995714323726567818, −6.15814047794797518466247762740, −4.44341645406027649200038875025, −2.19568258221450216734634513762, 0.19887850184247835096704003094, 1.21831551688700901783491181461, 3.30716397675569232821474390369, 5.46368356546174798862983366294, 6.13854263180255882285850425161, 8.062487174228664314839230080507, 9.655559712735681040868375261445, 10.70456811206095860713385733225, 11.15800937258386669040849953974, 12.54893052762025233244416397631

Graph of the ZZ-function along the critical line