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Properties

Label 2-13-13.9-c9-0-0
Degree 22
Conductor 1313
Sign −0.849+0.527i-0.849 + 0.527i
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  + (−6.02 + 10.4i)2-s + (−96.9 + 167. i)3-s + (183. + 317. i)4-s − 26.3·5-s + (−1.16e3 − 2.02e3i)6-s + (−1.34e3 − 2.33e3i)7-s − 1.05e4·8-s + (−8.94e3 − 1.54e4i)9-s + (158. − 274. i)10-s + (−2.39e3 + 4.15e3i)11-s − 7.10e4·12-s + (−3.04e4 − 9.83e4i)13-s + 3.24e4·14-s + (2.54e3 − 4.41e3i)15-s + (−3.01e4 + 5.21e4i)16-s + (1.26e5 + 2.18e5i)17-s + ⋯
L(s)  = 1  + (−0.266 + 0.461i)2-s + (−0.690 + 1.19i)3-s + (0.358 + 0.620i)4-s − 0.0188·5-s + (−0.367 − 0.637i)6-s + (−0.212 − 0.367i)7-s − 0.914·8-s + (−0.454 − 0.786i)9-s + (0.00501 − 0.00868i)10-s + (−0.0493 + 0.0854i)11-s − 0.989·12-s + (−0.296 − 0.955i)13-s + 0.226·14-s + (0.0130 − 0.0225i)15-s + (−0.114 + 0.198i)16-s + (0.366 + 0.634i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((−0.849+0.527i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((−0.849+0.527i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: −0.849+0.527i-0.849 + 0.527i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(9,⋅)\chi_{13} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), −0.849+0.527i)(2,\ 13,\ (\ :9/2),\ -0.849 + 0.527i)

Particular Values

L(5)L(5) ≈\approx 0.191949−0.672631i0.191949 - 0.672631i
L(12)L(\frac12) ≈\approx 0.191949−0.672631i0.191949 - 0.672631i
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)
bad13 1+(3.04e4+9.83e4i)T 1 + (3.04e4 + 9.83e4i)T
good2 1+(6.02−10.4i)T+(−256−443.i)T2 1 + (6.02 - 10.4i)T + (-256 - 443. i)T^{2}
3 1+(96.9−167.i)T+(−9.84e3−1.70e4i)T2 1 + (96.9 - 167. i)T + (-9.84e3 - 1.70e4i)T^{2}
5 1+26.3T+1.95e6T2 1 + 26.3T + 1.95e6T^{2}
7 1+(1.34e3+2.33e3i)T+(−2.01e7+3.49e7i)T2 1 + (1.34e3 + 2.33e3i)T + (-2.01e7 + 3.49e7i)T^{2}
11 1+(2.39e3−4.15e3i)T+(−1.17e9−2.04e9i)T2 1 + (2.39e3 - 4.15e3i)T + (-1.17e9 - 2.04e9i)T^{2}
17 1+(−1.26e5−2.18e5i)T+(−5.92e10+1.02e11i)T2 1 + (-1.26e5 - 2.18e5i)T + (-5.92e10 + 1.02e11i)T^{2}
19 1+(−2.78e5−4.82e5i)T+(−1.61e11+2.79e11i)T2 1 + (-2.78e5 - 4.82e5i)T + (-1.61e11 + 2.79e11i)T^{2}
23 1+(1.53e5−2.65e5i)T+(−9.00e11−1.55e12i)T2 1 + (1.53e5 - 2.65e5i)T + (-9.00e11 - 1.55e12i)T^{2}
29 1+(2.77e6−4.81e6i)T+(−7.25e12−1.25e13i)T2 1 + (2.77e6 - 4.81e6i)T + (-7.25e12 - 1.25e13i)T^{2}
31 1+3.02e6T+2.64e13T2 1 + 3.02e6T + 2.64e13T^{2}
37 1+(−7.14e6+1.23e7i)T+(−6.49e13−1.12e14i)T2 1 + (-7.14e6 + 1.23e7i)T + (-6.49e13 - 1.12e14i)T^{2}
41 1+(1.16e7−2.02e7i)T+(−1.63e14−2.83e14i)T2 1 + (1.16e7 - 2.02e7i)T + (-1.63e14 - 2.83e14i)T^{2}
43 1+(−2.14e7−3.72e7i)T+(−2.51e14+4.35e14i)T2 1 + (-2.14e7 - 3.72e7i)T + (-2.51e14 + 4.35e14i)T^{2}
47 1−2.42e7T+1.11e15T2 1 - 2.42e7T + 1.11e15T^{2}
53 1+6.28e7T+3.29e15T2 1 + 6.28e7T + 3.29e15T^{2}
59 1+(−4.66e7−8.08e7i)T+(−4.33e15+7.50e15i)T2 1 + (-4.66e7 - 8.08e7i)T + (-4.33e15 + 7.50e15i)T^{2}
61 1+(1.07e7+1.85e7i)T+(−5.84e15+1.01e16i)T2 1 + (1.07e7 + 1.85e7i)T + (-5.84e15 + 1.01e16i)T^{2}
67 1+(−9.99e7+1.73e8i)T+(−1.36e16−2.35e16i)T2 1 + (-9.99e7 + 1.73e8i)T + (-1.36e16 - 2.35e16i)T^{2}
71 1+(−1.27e8−2.21e8i)T+(−2.29e16+3.97e16i)T2 1 + (-1.27e8 - 2.21e8i)T + (-2.29e16 + 3.97e16i)T^{2}
73 1−1.64e7T+5.88e16T2 1 - 1.64e7T + 5.88e16T^{2}
79 1+1.39e8T+1.19e17T2 1 + 1.39e8T + 1.19e17T^{2}
83 1−3.64e8T+1.86e17T2 1 - 3.64e8T + 1.86e17T^{2}
89 1+(5.80e8−1.00e9i)T+(−1.75e17−3.03e17i)T2 1 + (5.80e8 - 1.00e9i)T + (-1.75e17 - 3.03e17i)T^{2}
97 1+(1.89e8+3.28e8i)T+(−3.80e17+6.58e17i)T2 1 + (1.89e8 + 3.28e8i)T + (-3.80e17 + 6.58e17i)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

−17.93229873147887871187333913508, −16.81464257238157003950958281504, −16.07694198473499614264121814872, −14.91883701002373592338017637452, −12.60217410584602126667615231198, −11.05492846770605494125020673963, −9.713227878557483332817002166355, −7.73711103259852922890177030063, −5.72062841336620563854103428193, −3.65862253641760444350475853061, 0.43992966011112630111781861369, 2.09545215114098101987733649320, 5.77961195924373777327709730125, 7.12426605224116315788092876028, 9.489341672810330661891562978611, 11.36048117568791163169312333074, 12.15751457529098082997917145255, 13.80485166480576551994953505593, 15.59140309722907087571147851093, 17.27599088573636237076833075433

Graph of the ZZ-function along the critical line