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Properties

Label 2-63-7.4-c5-0-10
Degree 22
Conductor 6363
Sign 0.994−0.102i0.994 - 0.102i
Analytic cond. 10.104110.1041
Root an. cond. 3.178703.17870
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 2.37i)2-s + (12.2 − 21.1i)4-s + (29.1 + 50.5i)5-s + (−21.4 − 127. i)7-s + 155.·8-s + (−80.2 + 138. i)10-s + (8.71 − 15.0i)11-s + 889.·13-s + (274. − 226. i)14-s + (−178. − 308. i)16-s + (513. − 889. i)17-s + (869. + 1.50e3i)19-s + 1.42e3·20-s + 47.8·22-s + (1.96e3 + 3.40e3i)23-s + ⋯
L(s)  = 1  + (0.242 + 0.420i)2-s + (0.381 − 0.661i)4-s + (0.522 + 0.904i)5-s + (−0.165 − 0.986i)7-s + 0.856·8-s + (−0.253 + 0.439i)10-s + (0.0217 − 0.0376i)11-s + 1.46·13-s + (0.374 − 0.309i)14-s + (−0.173 − 0.301i)16-s + (0.430 − 0.746i)17-s + (0.552 + 0.957i)19-s + 0.797·20-s + 0.0210·22-s + (0.775 + 1.34i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.994−0.102i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+5/2) L(s)=((0.994−0.102i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.994−0.102i0.994 - 0.102i
Analytic conductor: 10.104110.1041
Root analytic conductor: 3.178703.17870
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ63(46,⋅)\chi_{63} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :5/2), 0.994−0.102i)(2,\ 63,\ (\ :5/2),\ 0.994 - 0.102i)

Particular Values

L(3)L(3) ≈\approx 2.47001+0.126996i2.47001 + 0.126996i
L(12)L(\frac12) ≈\approx 2.47001+0.126996i2.47001 + 0.126996i
L(72)L(\frac{7}{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+(21.4+127.i)T 1 + (21.4 + 127. i)T
good2 1+(−1.37−2.37i)T+(−16+27.7i)T2 1 + (-1.37 - 2.37i)T + (-16 + 27.7i)T^{2}
5 1+(−29.1−50.5i)T+(−1.56e3+2.70e3i)T2 1 + (-29.1 - 50.5i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−8.71+15.0i)T+(−8.05e4−1.39e5i)T2 1 + (-8.71 + 15.0i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−889.T+3.71e5T2 1 - 889.T + 3.71e5T^{2}
17 1+(−513.+889.i)T+(−7.09e5−1.22e6i)T2 1 + (-513. + 889. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−869.−1.50e3i)T+(−1.23e6+2.14e6i)T2 1 + (-869. - 1.50e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−1.96e3−3.40e3i)T+(−3.21e6+5.57e6i)T2 1 + (-1.96e3 - 3.40e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+5.63e3T+2.05e7T2 1 + 5.63e3T + 2.05e7T^{2}
31 1+(−1.54e3+2.68e3i)T+(−1.43e7−2.47e7i)T2 1 + (-1.54e3 + 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(2.51e3+4.35e3i)T+(−3.46e7+6.00e7i)T2 1 + (2.51e3 + 4.35e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+1.83e4T+1.15e8T2 1 + 1.83e4T + 1.15e8T^{2}
43 1+1.63e3T+1.47e8T2 1 + 1.63e3T + 1.47e8T^{2}
47 1+(4.80e3+8.31e3i)T+(−1.14e8+1.98e8i)T2 1 + (4.80e3 + 8.31e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(1.16e4−2.01e4i)T+(−2.09e8−3.62e8i)T2 1 + (1.16e4 - 2.01e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.80e3−3.12e3i)T+(−3.57e8−6.19e8i)T2 1 + (1.80e3 - 3.12e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(1.14e4+1.98e4i)T+(−4.22e8+7.31e8i)T2 1 + (1.14e4 + 1.98e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.35e4−4.07e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.35e4 - 4.07e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−1.59e3T+1.80e9T2 1 - 1.59e3T + 1.80e9T^{2}
73 1+(2.96e3−5.13e3i)T+(−1.03e9−1.79e9i)T2 1 + (2.96e3 - 5.13e3i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−4.42e4−7.66e4i)T+(−1.53e9+2.66e9i)T2 1 + (-4.42e4 - 7.66e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−9.58e4T+3.93e9T2 1 - 9.58e4T + 3.93e9T^{2}
89 1+(2.32e4+4.02e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+7.59e4T+8.58e9T2 1 + 7.59e4T + 8.58e9T^{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

−13.91921655509214965507532250650, −13.48963636943716245794381606601, −11.39999928392880380861640886849, −10.57791415580590568498951403150, −9.644740063336668464470147278693, −7.58263478147098886454878801947, −6.60440072420327010918554362523, −5.51930212590954880588010106326, −3.51283288971617070275976505547, −1.38095645184881750988500101384, 1.60100842539679883761423261546, 3.25916490878785512029035979922, 5.02018787831344439120887918505, 6.46680667916689134201252454535, 8.302501559144770194127863087550, 9.105021111815970588157067465406, 10.76557484746689654554130852349, 11.88228823810789006660822442664, 12.85690485658925676281368137808, 13.44745199180996289892886574073

Graph of the ZZ-function along the critical line