Properties

Label 2-13-13.3-c9-0-7
Degree 22
Conductor 1313
Sign 0.461+0.886i0.461 + 0.886i
Analytic cond. 6.695466.69546
Root an. cond. 2.587552.58755
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  + (3.19 + 5.52i)2-s + (2.91 + 5.04i)3-s + (235. − 408. i)4-s − 1.06e3·5-s + (−18.5 + 32.2i)6-s + (3.23e3 − 5.60e3i)7-s + 6.27e3·8-s + (9.82e3 − 1.70e4i)9-s + (−3.39e3 − 5.88e3i)10-s + (−9.31e3 − 1.61e4i)11-s + 2.74e3·12-s + (7.63e4 − 6.90e4i)13-s + 4.13e4·14-s + (−3.09e3 − 5.36e3i)15-s + (−1.00e5 − 1.74e5i)16-s + (−2.60e5 + 4.52e5i)17-s + ⋯
L(s)  = 1  + (0.141 + 0.244i)2-s + (0.0207 + 0.0359i)3-s + (0.460 − 0.797i)4-s − 0.761·5-s + (−0.00585 + 0.0101i)6-s + (0.509 − 0.882i)7-s + 0.541·8-s + (0.499 − 0.864i)9-s + (−0.107 − 0.185i)10-s + (−0.191 − 0.332i)11-s + 0.0382·12-s + (0.741 − 0.670i)13-s + 0.287·14-s + (−0.0158 − 0.0273i)15-s + (−0.383 − 0.664i)16-s + (−0.757 + 1.31i)17-s + ⋯

Functional equation

Λ(s)=(13s/2ΓC(s)L(s)=((0.461+0.886i)Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2ΓC(s+9/2)L(s)=((0.461+0.886i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.461+0.886i0.461 + 0.886i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(3,)\chi_{13} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), 0.461+0.886i)(2,\ 13,\ (\ :9/2),\ 0.461 + 0.886i)

Particular Values

L(5)L(5) \approx 1.501810.911191i1.50181 - 0.911191i
L(12)L(\frac12) \approx 1.501810.911191i1.50181 - 0.911191i
L(112)L(\frac{11}{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)
bad13 1+(7.63e4+6.90e4i)T 1 + (-7.63e4 + 6.90e4i)T
good2 1+(3.195.52i)T+(256+443.i)T2 1 + (-3.19 - 5.52i)T + (-256 + 443. i)T^{2}
3 1+(2.915.04i)T+(9.84e3+1.70e4i)T2 1 + (-2.91 - 5.04i)T + (-9.84e3 + 1.70e4i)T^{2}
5 1+1.06e3T+1.95e6T2 1 + 1.06e3T + 1.95e6T^{2}
7 1+(3.23e3+5.60e3i)T+(2.01e73.49e7i)T2 1 + (-3.23e3 + 5.60e3i)T + (-2.01e7 - 3.49e7i)T^{2}
11 1+(9.31e3+1.61e4i)T+(1.17e9+2.04e9i)T2 1 + (9.31e3 + 1.61e4i)T + (-1.17e9 + 2.04e9i)T^{2}
17 1+(2.60e54.52e5i)T+(5.92e101.02e11i)T2 1 + (2.60e5 - 4.52e5i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(1.36e52.37e5i)T+(1.61e112.79e11i)T2 1 + (1.36e5 - 2.37e5i)T + (-1.61e11 - 2.79e11i)T^{2}
23 1+(4.42e57.67e5i)T+(9.00e11+1.55e12i)T2 1 + (-4.42e5 - 7.67e5i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+(1.89e6+3.27e6i)T+(7.25e12+1.25e13i)T2 1 + (1.89e6 + 3.27e6i)T + (-7.25e12 + 1.25e13i)T^{2}
31 19.87e6T+2.64e13T2 1 - 9.87e6T + 2.64e13T^{2}
37 1+(5.10e68.84e6i)T+(6.49e13+1.12e14i)T2 1 + (-5.10e6 - 8.84e6i)T + (-6.49e13 + 1.12e14i)T^{2}
41 1+(6.37e61.10e7i)T+(1.63e14+2.83e14i)T2 1 + (-6.37e6 - 1.10e7i)T + (-1.63e14 + 2.83e14i)T^{2}
43 1+(2.01e63.49e6i)T+(2.51e144.35e14i)T2 1 + (2.01e6 - 3.49e6i)T + (-2.51e14 - 4.35e14i)T^{2}
47 11.84e6T+1.11e15T2 1 - 1.84e6T + 1.11e15T^{2}
53 15.32e7T+3.29e15T2 1 - 5.32e7T + 3.29e15T^{2}
59 1+(8.32e6+1.44e7i)T+(4.33e157.50e15i)T2 1 + (-8.32e6 + 1.44e7i)T + (-4.33e15 - 7.50e15i)T^{2}
61 1+(4.50e7+7.79e7i)T+(5.84e151.01e16i)T2 1 + (-4.50e7 + 7.79e7i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(1.30e7+2.25e7i)T+(1.36e16+2.35e16i)T2 1 + (1.30e7 + 2.25e7i)T + (-1.36e16 + 2.35e16i)T^{2}
71 1+(1.40e82.42e8i)T+(2.29e163.97e16i)T2 1 + (1.40e8 - 2.42e8i)T + (-2.29e16 - 3.97e16i)T^{2}
73 11.22e8T+5.88e16T2 1 - 1.22e8T + 5.88e16T^{2}
79 1+4.73e8T+1.19e17T2 1 + 4.73e8T + 1.19e17T^{2}
83 17.61e8T+1.86e17T2 1 - 7.61e8T + 1.86e17T^{2}
89 1+(6.13e7+1.06e8i)T+(1.75e17+3.03e17i)T2 1 + (6.13e7 + 1.06e8i)T + (-1.75e17 + 3.03e17i)T^{2}
97 1+(1.54e82.67e8i)T+(3.80e176.58e17i)T2 1 + (1.54e8 - 2.67e8i)T + (-3.80e17 - 6.58e17i)T^{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

−17.38739583197867975202272254005, −15.71415516789454939450454471482, −14.99465133376969635252997285163, −13.38237422398847672088447366298, −11.43615967380529443095532286541, −10.25782641757604214318167310958, −7.961279999318975844122005836111, −6.30130408858647237033894471436, −4.09573368744352287510834433970, −1.02391604874757114450532487375, 2.34329966717100102972544402589, 4.49428699417159863637617750974, 7.20719384804755308420776295939, 8.590412995038826874338153360985, 11.07210209192433145988525756964, 12.02284421561984009502331059125, 13.47652568288825034729775560652, 15.48066865405650025967289282079, 16.30418223244241988854972383950, 18.04288638886111515186374722593

Graph of the ZZ-function along the critical line