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Properties

Label 2-63-7.4-c9-0-11
Degree 22
Conductor 6363
Sign −0.839−0.542i-0.839 - 0.542i
Analytic cond. 32.447232.4472
Root an. cond. 5.696245.69624
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 + 27.5i)2-s + (−250. + 434. i)4-s + (502. + 870. i)5-s + (3.10e3 − 5.54e3i)7-s + 332.·8-s + (−1.60e4 + 2.77e4i)10-s + (−3.79e4 + 6.57e4i)11-s + 1.66e5·13-s + (2.02e5 − 2.70e3i)14-s + (1.33e5 + 2.31e5i)16-s + (−2.17e5 + 3.76e5i)17-s + (−2.24e5 − 3.88e5i)19-s − 5.04e5·20-s − 2.41e6·22-s + (8.21e5 + 1.42e6i)23-s + ⋯
L(s)  = 1  + (0.703 + 1.21i)2-s + (−0.489 + 0.848i)4-s + (0.359 + 0.622i)5-s + (0.488 − 0.872i)7-s + 0.0286·8-s + (−0.505 + 0.876i)10-s + (−0.781 + 1.35i)11-s + 1.61·13-s + (1.40 − 0.0188i)14-s + (0.509 + 0.883i)16-s + (−0.630 + 1.09i)17-s + (−0.395 − 0.684i)19-s − 0.704·20-s − 2.19·22-s + (0.611 + 1.05i)23-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.839−0.542i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+9/2) L(s)=((−0.839−0.542i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.839−0.542i-0.839 - 0.542i
Analytic conductor: 32.447232.4472
Root analytic conductor: 5.696245.69624
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ63(46,⋅)\chi_{63} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :9/2), −0.839−0.542i)(2,\ 63,\ (\ :9/2),\ -0.839 - 0.542i)

Particular Values

L(5)L(5) ≈\approx 0.933940+3.16666i0.933940 + 3.16666i
L(12)L(\frac12) ≈\approx 0.933940+3.16666i0.933940 + 3.16666i
L(112)L(\frac{11}{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+(−3.10e3+5.54e3i)T 1 + (-3.10e3 + 5.54e3i)T
good2 1+(−15.9−27.5i)T+(−256+443.i)T2 1 + (-15.9 - 27.5i)T + (-256 + 443. i)T^{2}
5 1+(−502.−870.i)T+(−9.76e5+1.69e6i)T2 1 + (-502. - 870. i)T + (-9.76e5 + 1.69e6i)T^{2}
11 1+(3.79e4−6.57e4i)T+(−1.17e9−2.04e9i)T2 1 + (3.79e4 - 6.57e4i)T + (-1.17e9 - 2.04e9i)T^{2}
13 1−1.66e5T+1.06e10T2 1 - 1.66e5T + 1.06e10T^{2}
17 1+(2.17e5−3.76e5i)T+(−5.92e10−1.02e11i)T2 1 + (2.17e5 - 3.76e5i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(2.24e5+3.88e5i)T+(−1.61e11+2.79e11i)T2 1 + (2.24e5 + 3.88e5i)T + (-1.61e11 + 2.79e11i)T^{2}
23 1+(−8.21e5−1.42e6i)T+(−9.00e11+1.55e12i)T2 1 + (-8.21e5 - 1.42e6i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+3.82e6T+1.45e13T2 1 + 3.82e6T + 1.45e13T^{2}
31 1+(8.39e5−1.45e6i)T+(−1.32e13−2.28e13i)T2 1 + (8.39e5 - 1.45e6i)T + (-1.32e13 - 2.28e13i)T^{2}
37 1+(−6.88e6−1.19e7i)T+(−6.49e13+1.12e14i)T2 1 + (-6.88e6 - 1.19e7i)T + (-6.49e13 + 1.12e14i)T^{2}
41 1−6.02e5T+3.27e14T2 1 - 6.02e5T + 3.27e14T^{2}
43 1+2.34e7T+5.02e14T2 1 + 2.34e7T + 5.02e14T^{2}
47 1+(2.08e7+3.60e7i)T+(−5.59e14+9.69e14i)T2 1 + (2.08e7 + 3.60e7i)T + (-5.59e14 + 9.69e14i)T^{2}
53 1+(−2.05e7+3.56e7i)T+(−1.64e15−2.85e15i)T2 1 + (-2.05e7 + 3.56e7i)T + (-1.64e15 - 2.85e15i)T^{2}
59 1+(2.32e7−4.02e7i)T+(−4.33e15−7.50e15i)T2 1 + (2.32e7 - 4.02e7i)T + (-4.33e15 - 7.50e15i)T^{2}
61 1+(−8.99e7−1.55e8i)T+(−5.84e15+1.01e16i)T2 1 + (-8.99e7 - 1.55e8i)T + (-5.84e15 + 1.01e16i)T^{2}
67 1+(5.56e7−9.64e7i)T+(−1.36e16−2.35e16i)T2 1 + (5.56e7 - 9.64e7i)T + (-1.36e16 - 2.35e16i)T^{2}
71 1−1.92e8T+4.58e16T2 1 - 1.92e8T + 4.58e16T^{2}
73 1+(−1.82e8+3.15e8i)T+(−2.94e16−5.09e16i)T2 1 + (-1.82e8 + 3.15e8i)T + (-2.94e16 - 5.09e16i)T^{2}
79 1+(1.68e7+2.91e7i)T+(−5.99e16+1.03e17i)T2 1 + (1.68e7 + 2.91e7i)T + (-5.99e16 + 1.03e17i)T^{2}
83 1−2.52e8T+1.86e17T2 1 - 2.52e8T + 1.86e17T^{2}
89 1+(−3.67e8−6.36e8i)T+(−1.75e17+3.03e17i)T2 1 + (-3.67e8 - 6.36e8i)T + (-1.75e17 + 3.03e17i)T^{2}
97 1−4.21e7T+7.60e17T2 1 - 4.21e7T + 7.60e17T^{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

−13.44656248308386126169982232715, −13.20039002419987091871272267701, −11.08091130188965020733351616324, −10.26395274386050082295087375991, −8.362329008578001792239879050499, −7.21311771133954201314193209687, −6.39525127900641999893699542652, −5.01107509621570934476162553667, −3.84256189941213004713636059086, −1.69671932003328070679797297740, 0.827268061713329168778232990168, 2.13742216108495019076655249125, 3.38913231559792340693660649800, 4.91714410669609246312389915181, 5.90575903508278056654561598570, 8.223143016077723730318800795078, 9.196420082102448773836500861854, 10.93573672708974484779153768621, 11.30628062566132135547380333762, 12.74181451523957222780400473078

Graph of the ZZ-function along the critical line