Properties

Label 2-63-21.17-c7-0-10
Degree 22
Conductor 6363
Sign 0.982+0.187i0.982 + 0.187i
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  + (−18.8 + 10.8i)2-s + (172. − 297. i)4-s + (38.2 + 66.1i)5-s + (884. + 201. i)7-s + 4.69e3i·8-s + (−1.43e3 − 830. i)10-s + (−338. − 195. i)11-s − 9.29e3i·13-s + (−1.88e4 + 5.82e3i)14-s + (−2.89e4 − 5.01e4i)16-s + (1.01e4 − 1.75e4i)17-s + (−4.12e4 + 2.38e4i)19-s + 2.62e4·20-s + 8.48e3·22-s + (−5.30e4 + 3.06e4i)23-s + ⋯
L(s)  = 1  + (−1.66 + 0.960i)2-s + (1.34 − 2.32i)4-s + (0.136 + 0.236i)5-s + (0.975 + 0.221i)7-s + 3.24i·8-s + (−0.454 − 0.262i)10-s + (−0.0766 − 0.0442i)11-s − 1.17i·13-s + (−1.83 + 0.567i)14-s + (−1.76 − 3.06i)16-s + (0.501 − 0.868i)17-s + (−1.37 + 0.796i)19-s + 0.734·20-s + 0.169·22-s + (−0.908 + 0.524i)23-s + ⋯

Functional equation

Λ(s)=(63s/2ΓC(s)L(s)=((0.982+0.187i)Λ(8s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2ΓC(s+7/2)L(s)=((0.982+0.187i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    3273^{2} \cdot 7
Sign: 0.982+0.187i0.982 + 0.187i
Analytic conductor: 19.680219.6802
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ63(17,)\chi_{63} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :7/2), 0.982+0.187i)(2,\ 63,\ (\ :7/2),\ 0.982 + 0.187i)

Particular Values

L(4)L(4) \approx 0.7809560.0740053i0.780956 - 0.0740053i
L(12)L(\frac12) \approx 0.7809560.0740053i0.780956 - 0.0740053i
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1+(884.201.i)T 1 + (-884. - 201. i)T
good2 1+(18.810.8i)T+(64110.i)T2 1 + (18.8 - 10.8i)T + (64 - 110. i)T^{2}
5 1+(38.266.1i)T+(3.90e4+6.76e4i)T2 1 + (-38.2 - 66.1i)T + (-3.90e4 + 6.76e4i)T^{2}
11 1+(338.+195.i)T+(9.74e6+1.68e7i)T2 1 + (338. + 195. i)T + (9.74e6 + 1.68e7i)T^{2}
13 1+9.29e3iT6.27e7T2 1 + 9.29e3iT - 6.27e7T^{2}
17 1+(1.01e4+1.75e4i)T+(2.05e83.55e8i)T2 1 + (-1.01e4 + 1.75e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(4.12e42.38e4i)T+(4.46e87.74e8i)T2 1 + (4.12e4 - 2.38e4i)T + (4.46e8 - 7.74e8i)T^{2}
23 1+(5.30e43.06e4i)T+(1.70e92.94e9i)T2 1 + (5.30e4 - 3.06e4i)T + (1.70e9 - 2.94e9i)T^{2}
29 1+1.36e5iT1.72e10T2 1 + 1.36e5iT - 1.72e10T^{2}
31 1+(2.25e51.30e5i)T+(1.37e10+2.38e10i)T2 1 + (-2.25e5 - 1.30e5i)T + (1.37e10 + 2.38e10i)T^{2}
37 1+(4.99e4+8.65e4i)T+(4.74e10+8.22e10i)T2 1 + (4.99e4 + 8.65e4i)T + (-4.74e10 + 8.22e10i)T^{2}
41 1+5.63e4T+1.94e11T2 1 + 5.63e4T + 1.94e11T^{2}
43 11.68e4T+2.71e11T2 1 - 1.68e4T + 2.71e11T^{2}
47 1+(6.38e5+1.10e6i)T+(2.53e11+4.38e11i)T2 1 + (6.38e5 + 1.10e6i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(8.22e54.74e5i)T+(5.87e11+1.01e12i)T2 1 + (-8.22e5 - 4.74e5i)T + (5.87e11 + 1.01e12i)T^{2}
59 1+(6.91e5+1.19e6i)T+(1.24e122.15e12i)T2 1 + (-6.91e5 + 1.19e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(2.50e6+1.44e6i)T+(1.57e122.72e12i)T2 1 + (-2.50e6 + 1.44e6i)T + (1.57e12 - 2.72e12i)T^{2}
67 1+(1.28e6+2.21e6i)T+(3.03e125.24e12i)T2 1 + (-1.28e6 + 2.21e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+4.17e6iT9.09e12T2 1 + 4.17e6iT - 9.09e12T^{2}
73 1+(7.16e5+4.13e5i)T+(5.52e12+9.56e12i)T2 1 + (7.16e5 + 4.13e5i)T + (5.52e12 + 9.56e12i)T^{2}
79 1+(2.32e64.02e6i)T+(9.60e12+1.66e13i)T2 1 + (-2.32e6 - 4.02e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 16.11e6T+2.71e13T2 1 - 6.11e6T + 2.71e13T^{2}
89 1+(1.09e6+1.90e6i)T+(2.21e13+3.83e13i)T2 1 + (1.09e6 + 1.90e6i)T + (-2.21e13 + 3.83e13i)T^{2}
97 1+3.26e6iT8.07e13T2 1 + 3.26e6iT - 8.07e13T^{2}
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   L(s)=p j=12(1αj,pps)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

−14.04418374632796369761416427115, −11.90330383202100227708819563629, −10.62807131962521145010290247271, −9.923396095348971651729065090184, −8.395537435699786318367622575628, −7.902832545589091251747881300777, −6.45388415086579962974953986896, −5.27834488388230950513279666913, −2.12298866478104845591365591478, −0.56559037413349577110976035761, 1.16863305820783200958322172542, 2.25734392161315708666424338642, 4.19991223373421822858691644502, 6.78714460448712760208400692358, 8.140431447549642904750027028544, 8.862838104758773746392277133336, 10.12797825306932529550471362059, 11.05403625976599338762787417425, 11.91944968819872576665433149626, 13.07230995752348330670605316839

Graph of the ZZ-function along the critical line