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Properties

Label 2-63-7.4-c3-0-8
Degree 22
Conductor 6363
Sign −0.502−0.864i-0.502 - 0.864i
Analytic cond. 3.717123.71712
Root an. cond. 1.927981.92798
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.02 − 3.51i)2-s + (−4.22 + 7.31i)4-s + (−4.96 − 8.59i)5-s + (−15.3 + 10.2i)7-s + 1.80·8-s + (−20.1 + 34.8i)10-s + (−6.76 + 11.7i)11-s + 18.5·13-s + (67.3 + 33.1i)14-s + (30.1 + 52.1i)16-s + (−46.8 + 81.1i)17-s + (−65.9 − 114. i)19-s + 83.7·20-s + 54.9·22-s + (−99.1 − 171. i)23-s + ⋯
L(s)  = 1  + (−0.716 − 1.24i)2-s + (−0.527 + 0.914i)4-s + (−0.443 − 0.768i)5-s + (−0.831 + 0.556i)7-s + 0.0799·8-s + (−0.636 + 1.10i)10-s + (−0.185 + 0.321i)11-s + 0.395·13-s + (1.28 + 0.633i)14-s + (0.470 + 0.815i)16-s + (−0.668 + 1.15i)17-s + (−0.796 − 1.37i)19-s + 0.936·20-s + 0.532·22-s + (−0.898 − 1.55i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.502−0.864i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+3/2) L(s)=((−0.502−0.864i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.502−0.864i-0.502 - 0.864i
Analytic conductor: 3.717123.71712
Root analytic conductor: 1.927981.92798
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ63(46,⋅)\chi_{63} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :3/2), −0.502−0.864i)(2,\ 63,\ (\ :3/2),\ -0.502 - 0.864i)

Particular Values

L(2)L(2) ≈\approx 0.121917+0.211830i0.121917 + 0.211830i
L(12)L(\frac12) ≈\approx 0.121917+0.211830i0.121917 + 0.211830i
L(52)L(\frac{5}{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+(15.3−10.2i)T 1 + (15.3 - 10.2i)T
good2 1+(2.02+3.51i)T+(−4+6.92i)T2 1 + (2.02 + 3.51i)T + (-4 + 6.92i)T^{2}
5 1+(4.96+8.59i)T+(−62.5+108.i)T2 1 + (4.96 + 8.59i)T + (-62.5 + 108. i)T^{2}
11 1+(6.76−11.7i)T+(−665.5−1.15e3i)T2 1 + (6.76 - 11.7i)T + (-665.5 - 1.15e3i)T^{2}
13 1−18.5T+2.19e3T2 1 - 18.5T + 2.19e3T^{2}
17 1+(46.8−81.1i)T+(−2.45e3−4.25e3i)T2 1 + (46.8 - 81.1i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(65.9+114.i)T+(−3.42e3+5.94e3i)T2 1 + (65.9 + 114. i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(99.1+171.i)T+(−6.08e3+1.05e4i)T2 1 + (99.1 + 171. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+188.T+2.43e4T2 1 + 188.T + 2.43e4T^{2}
31 1+(−41.9+72.6i)T+(−1.48e4−2.57e4i)T2 1 + (-41.9 + 72.6i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(40.0+69.4i)T+(−2.53e4+4.38e4i)T2 1 + (40.0 + 69.4i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−385.T+6.89e4T2 1 - 385.T + 6.89e4T^{2}
43 1+397.T+7.95e4T2 1 + 397.T + 7.95e4T^{2}
47 1+(−136.−235.i)T+(−5.19e4+8.99e4i)T2 1 + (-136. - 235. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(18.4−32.0i)T+(−7.44e4−1.28e5i)T2 1 + (18.4 - 32.0i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−197.+342.i)T+(−1.02e5−1.77e5i)T2 1 + (-197. + 342. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(6.73+11.6i)T+(−1.13e5+1.96e5i)T2 1 + (6.73 + 11.6i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(170.−294.i)T+(−1.50e5−2.60e5i)T2 1 + (170. - 294. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−211.T+3.57e5T2 1 - 211.T + 3.57e5T^{2}
73 1+(243.−420.i)T+(−1.94e5−3.36e5i)T2 1 + (243. - 420. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(146.+254.i)T+(−2.46e5+4.26e5i)T2 1 + (146. + 254. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+889.T+5.71e5T2 1 + 889.T + 5.71e5T^{2}
89 1+(−572.−991.i)T+(−3.52e5+6.10e5i)T2 1 + (-572. - 991. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−1.38e3T+9.12e5T2 1 - 1.38e3T + 9.12e5T^{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

−12.92526650637352653425148796557, −12.62090867009729020666176706042, −11.36218725815393114791066713160, −10.32365868375636226618318831984, −9.087162956167776427077875260870, −8.405850679509479994543582591383, −6.27759854454298615848025521097, −4.16079453089427303410431204321, −2.34415855697655614374052881179, −0.20086376823137974777784405416, 3.51422044183024914902467371682, 5.88428862609934178753440745886, 6.97555830947702637776027866577, 7.82535496348198660611732035375, 9.217775467864867395277422385374, 10.34533317331687338839195578366, 11.68447718374594030103337769954, 13.32835903254211411786012015213, 14.43466423540980467507913519277, 15.51534352490788719004066879019

Graph of the ZZ-function along the critical line