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Properties

Label 2-17-17.8-c11-0-2
Degree 22
Conductor 1717
Sign −0.997+0.0758i-0.997 + 0.0758i
Analytic cond. 13.061813.0618
Root an. cond. 3.614113.61411
Motivic weight 1111
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.6 − 18.6i)2-s + (−274. + 663. i)3-s + 1.34e3i·4-s + (1.89e3 + 783. i)5-s + (7.26e3 + 1.75e4i)6-s + (−1.94e4 + 8.05e3i)7-s + (6.35e4 + 6.35e4i)8-s + (−2.39e5 − 2.39e5i)9-s + (5.00e4 − 2.07e4i)10-s + (−2.12e5 − 5.13e5i)11-s + (−8.95e5 − 3.70e5i)12-s + 7.21e4i·13-s + (−2.13e5 + 5.14e5i)14-s + (−1.03e6 + 1.03e6i)15-s − 3.88e5·16-s + (55.2 + 5.85e6i)17-s + ⋯
L(s)  = 1  + (0.413 − 0.413i)2-s + (−0.652 + 1.57i)3-s + 0.658i·4-s + (0.270 + 0.112i)5-s + (0.381 + 0.920i)6-s + (−0.437 + 0.181i)7-s + (0.685 + 0.685i)8-s + (−1.35 − 1.35i)9-s + (0.158 − 0.0655i)10-s + (−0.398 − 0.961i)11-s + (−1.03 − 0.430i)12-s + 0.0539i·13-s + (−0.105 + 0.255i)14-s + (−0.353 + 0.353i)15-s − 0.0926·16-s + (9.43e−6 + 0.999i)17-s + ⋯

Functional equation

Λ(s)=(17s/2 ΓC(s) L(s)=((−0.997+0.0758i) Λ‾(12−s)\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0758i)\, \overline{\Lambda}(12-s) \end{aligned}
Λ(s)=(17s/2 ΓC(s+11/2) L(s)=((−0.997+0.0758i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1717
Sign: −0.997+0.0758i-0.997 + 0.0758i
Analytic conductor: 13.061813.0618
Root analytic conductor: 3.614113.61411
Motivic weight: 1111
Rational: no
Arithmetic: yes
Character: χ17(8,⋅)\chi_{17} (8, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 17, ( :11/2), −0.997+0.0758i)(2,\ 17,\ (\ :11/2),\ -0.997 + 0.0758i)

Particular Values

L(6)L(6) ≈\approx 0.0395133−1.03975i0.0395133 - 1.03975i
L(12)L(\frac12) ≈\approx 0.0395133−1.03975i0.0395133 - 1.03975i
L(132)L(\frac{13}{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)
bad17 1+(−55.2−5.85e6i)T 1 + (-55.2 - 5.85e6i)T
good2 1+(−18.6+18.6i)T−2.04e3iT2 1 + (-18.6 + 18.6i)T - 2.04e3iT^{2}
3 1+(274.−663.i)T+(−1.25e5−1.25e5i)T2 1 + (274. - 663. i)T + (-1.25e5 - 1.25e5i)T^{2}
5 1+(−1.89e3−783.i)T+(3.45e7+3.45e7i)T2 1 + (-1.89e3 - 783. i)T + (3.45e7 + 3.45e7i)T^{2}
7 1+(1.94e4−8.05e3i)T+(1.39e9−1.39e9i)T2 1 + (1.94e4 - 8.05e3i)T + (1.39e9 - 1.39e9i)T^{2}
11 1+(2.12e5+5.13e5i)T+(−2.01e11+2.01e11i)T2 1 + (2.12e5 + 5.13e5i)T + (-2.01e11 + 2.01e11i)T^{2}
13 1−7.21e4iT−1.79e12T2 1 - 7.21e4iT - 1.79e12T^{2}
19 1+(8.42e5−8.42e5i)T−1.16e14iT2 1 + (8.42e5 - 8.42e5i)T - 1.16e14iT^{2}
23 1+(6.36e6+1.53e7i)T+(−6.73e14+6.73e14i)T2 1 + (6.36e6 + 1.53e7i)T + (-6.73e14 + 6.73e14i)T^{2}
29 1+(−1.14e8−4.74e7i)T+(8.62e15+8.62e15i)T2 1 + (-1.14e8 - 4.74e7i)T + (8.62e15 + 8.62e15i)T^{2}
31 1+(6.49e7−1.56e8i)T+(−1.79e16−1.79e16i)T2 1 + (6.49e7 - 1.56e8i)T + (-1.79e16 - 1.79e16i)T^{2}
37 1+(−2.51e8+6.06e8i)T+(−1.25e17−1.25e17i)T2 1 + (-2.51e8 + 6.06e8i)T + (-1.25e17 - 1.25e17i)T^{2}
41 1+(1.08e9−4.47e8i)T+(3.89e17−3.89e17i)T2 1 + (1.08e9 - 4.47e8i)T + (3.89e17 - 3.89e17i)T^{2}
43 1+(−7.28e8−7.28e8i)T+9.29e17iT2 1 + (-7.28e8 - 7.28e8i)T + 9.29e17iT^{2}
47 1−1.52e9iT−2.47e18T2 1 - 1.52e9iT - 2.47e18T^{2}
53 1+(6.14e7−6.14e7i)T−9.26e18iT2 1 + (6.14e7 - 6.14e7i)T - 9.26e18iT^{2}
59 1+(−5.34e9−5.34e9i)T+3.01e19iT2 1 + (-5.34e9 - 5.34e9i)T + 3.01e19iT^{2}
61 1+(8.61e9−3.56e9i)T+(3.07e19−3.07e19i)T2 1 + (8.61e9 - 3.56e9i)T + (3.07e19 - 3.07e19i)T^{2}
67 1−9.16e9T+1.22e20T2 1 - 9.16e9T + 1.22e20T^{2}
71 1+(7.19e9−1.73e10i)T+(−1.63e20−1.63e20i)T2 1 + (7.19e9 - 1.73e10i)T + (-1.63e20 - 1.63e20i)T^{2}
73 1+(6.96e9+2.88e9i)T+(2.21e20+2.21e20i)T2 1 + (6.96e9 + 2.88e9i)T + (2.21e20 + 2.21e20i)T^{2}
79 1+(−5.71e8−1.38e9i)T+(−5.28e20+5.28e20i)T2 1 + (-5.71e8 - 1.38e9i)T + (-5.28e20 + 5.28e20i)T^{2}
83 1+(4.11e10−4.11e10i)T−1.28e21iT2 1 + (4.11e10 - 4.11e10i)T - 1.28e21iT^{2}
89 1+7.13e10iT−2.77e21T2 1 + 7.13e10iT - 2.77e21T^{2}
97 1+(1.07e11+4.46e10i)T+(5.05e21+5.05e21i)T2 1 + (1.07e11 + 4.46e10i)T + (5.05e21 + 5.05e21i)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

−16.57577734125648911426149811846, −15.88683361364560749025975972923, −14.25116929467510048186409152424, −12.60767823535101623440292056165, −11.21112805655233519260312807289, −10.24573782099057873898320153871, −8.603053285495085487771605128139, −5.89855448545637745890470616632, −4.34842658026822725533590912496, −3.07512005370217102347159550499, 0.40595011660416146554438920670, 1.89025526411099286678454561894, 5.18671651628577487271322447669, 6.46626061754489918142680643513, 7.46607450865417350851190921623, 9.937744144328234760183694810107, 11.71106245689108575070813402704, 13.08613090360479337036950690657, 13.78226013820324240423416312519, 15.46231840747114655590134218887

Graph of the ZZ-function along the critical line