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Properties

Label 2-63-9.4-c7-0-2
Degree 22
Conductor 6363
Sign 0.999−0.0141i0.999 - 0.0141i
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  + (−8.75 − 15.1i)2-s + (−30.3 − 35.6i)3-s + (−89.2 + 154. i)4-s + (242. − 419. i)5-s + (−274. + 771. i)6-s + (−171.5 − 297. i)7-s + 885.·8-s + (−349. + 2.15e3i)9-s − 8.48e3·10-s + (779. + 1.34e3i)11-s + (8.21e3 − 1.50e3i)12-s + (−6.43e3 + 1.11e4i)13-s + (−3.00e3 + 5.20e3i)14-s + (−2.22e4 + 4.09e3i)15-s + (3.67e3 + 6.36e3i)16-s + 1.35e4·17-s + ⋯
L(s)  = 1  + (−0.773 − 1.34i)2-s + (−0.648 − 0.761i)3-s + (−0.697 + 1.20i)4-s + (0.867 − 1.50i)5-s + (−0.519 + 1.45i)6-s + (−0.188 − 0.327i)7-s + 0.611·8-s + (−0.159 + 0.987i)9-s − 2.68·10-s + (0.176 + 0.305i)11-s + (1.37 − 0.251i)12-s + (−0.812 + 1.40i)13-s + (−0.292 + 0.506i)14-s + (−1.70 + 0.313i)15-s + (0.224 + 0.388i)16-s + 0.668·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.999−0.0141i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+7/2) L(s)=((0.999−0.0141i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.999−0.0141i0.999 - 0.0141i
Analytic conductor: 19.680219.6802
Root analytic conductor: 4.436244.43624
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ63(22,⋅)\chi_{63} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :7/2), 0.999−0.0141i)(2,\ 63,\ (\ :7/2),\ 0.999 - 0.0141i)

Particular Values

L(4)L(4) ≈\approx 0.110277+0.000778227i0.110277 + 0.000778227i
L(12)L(\frac12) ≈\approx 0.110277+0.000778227i0.110277 + 0.000778227i
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+(30.3+35.6i)T 1 + (30.3 + 35.6i)T
7 1+(171.5+297.i)T 1 + (171.5 + 297. i)T
good2 1+(8.75+15.1i)T+(−64+110.i)T2 1 + (8.75 + 15.1i)T + (-64 + 110. i)T^{2}
5 1+(−242.+419.i)T+(−3.90e4−6.76e4i)T2 1 + (-242. + 419. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−779.−1.34e3i)T+(−9.74e6+1.68e7i)T2 1 + (-779. - 1.34e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(6.43e3−1.11e4i)T+(−3.13e7−5.43e7i)T2 1 + (6.43e3 - 1.11e4i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1−1.35e4T+4.10e8T2 1 - 1.35e4T + 4.10e8T^{2}
19 1+1.81e4T+8.93e8T2 1 + 1.81e4T + 8.93e8T^{2}
23 1+(4.20e4−7.28e4i)T+(−1.70e9−2.94e9i)T2 1 + (4.20e4 - 7.28e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(5.28e4+9.15e4i)T+(−8.62e9+1.49e10i)T2 1 + (5.28e4 + 9.15e4i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(−6.86e4+1.18e5i)T+(−1.37e10−2.38e10i)T2 1 + (-6.86e4 + 1.18e5i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1+2.44e5T+9.49e10T2 1 + 2.44e5T + 9.49e10T^{2}
41 1+(9.97e4−1.72e5i)T+(−9.73e10−1.68e11i)T2 1 + (9.97e4 - 1.72e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(−4.05e5−7.01e5i)T+(−1.35e11+2.35e11i)T2 1 + (-4.05e5 - 7.01e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(−2.97e5−5.14e5i)T+(−2.53e11+4.38e11i)T2 1 + (-2.97e5 - 5.14e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+1.84e6T+1.17e12T2 1 + 1.84e6T + 1.17e12T^{2}
59 1+(−8.24e5+1.42e6i)T+(−1.24e12−2.15e12i)T2 1 + (-8.24e5 + 1.42e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(9.91e5+1.71e6i)T+(−1.57e12+2.72e12i)T2 1 + (9.91e5 + 1.71e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(2.68e5−4.65e5i)T+(−3.03e12−5.24e12i)T2 1 + (2.68e5 - 4.65e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+4.85e6T+9.09e12T2 1 + 4.85e6T + 9.09e12T^{2}
73 1+1.55e6T+1.10e13T2 1 + 1.55e6T + 1.10e13T^{2}
79 1+(9.15e5+1.58e6i)T+(−9.60e12+1.66e13i)T2 1 + (9.15e5 + 1.58e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(−2.86e6−4.96e6i)T+(−1.35e13+2.35e13i)T2 1 + (-2.86e6 - 4.96e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1+3.26e6T+4.42e13T2 1 + 3.26e6T + 4.42e13T^{2}
97 1+(2.02e5+3.50e5i)T+(−4.03e13+6.99e13i)T2 1 + (2.02e5 + 3.50e5i)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

−12.95252378541249433988149279015, −12.24317742378459027831216014848, −11.43743313292330210766575966771, −9.900167296259189186433893871276, −9.300520833294399554298761509231, −7.891073990937244362612861919993, −6.06958730060397181918010321830, −4.51927427168953276214589666188, −2.02653537765451099242302725465, −1.29212961748470927660271159140, 0.05902122115011443176972337194, 3.01192611828824422347616387509, 5.46089958404344092160101573983, 6.19624477322929972898990540493, 7.22541105820751837540663464882, 8.820238803516563546455561031098, 10.18260672316855929036644622074, 10.44409170974104863833654610650, 12.26803802092806251014819239202, 14.25219248641726789751685786250

Graph of the ZZ-function along the critical line