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Properties

Label 2-34-17.12-c6-0-7
Degree 22
Conductor 3434
Sign 0.999+0.00333i0.999 + 0.00333i
Analytic cond. 7.821837.82183
Root an. cond. 2.796752.79675
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16 + 5.22i)2-s + (41.5 − 27.7i)3-s + (−22.6 + 22.6i)4-s + (53.2 + 10.5i)5-s + (234. + 156. i)6-s + (90.3 − 17.9i)7-s + (−167. − 69.2i)8-s + (675. − 1.63e3i)9-s + (59.8 + 301. i)10-s + (−4.84 + 7.24i)11-s + (−311. + 1.56e3i)12-s + (−39.4 − 39.4i)13-s + (289. + 433. i)14-s + (2.50e3 − 1.03e3i)15-s − 1.02e3i·16-s + (2.53e3 + 4.20e3i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (1.53 − 1.02i)3-s + (−0.353 + 0.353i)4-s + (0.425 + 0.0846i)5-s + (1.08 + 0.726i)6-s + (0.263 − 0.0524i)7-s + (−0.326 − 0.135i)8-s + (0.926 − 2.23i)9-s + (0.0598 + 0.301i)10-s + (−0.00363 + 0.00544i)11-s + (−0.180 + 0.907i)12-s + (−0.0179 − 0.0179i)13-s + (0.105 + 0.157i)14-s + (0.741 − 0.307i)15-s − 0.250i·16-s + (0.515 + 0.856i)17-s + ⋯

Functional equation

Λ(s)=(34s/2 ΓC(s) L(s)=((0.999+0.00333i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00333i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(34s/2 ΓC(s+3) L(s)=((0.999+0.00333i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.00333i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3434    =    2⋅172 \cdot 17
Sign: 0.999+0.00333i0.999 + 0.00333i
Analytic conductor: 7.821837.82183
Root analytic conductor: 2.796752.79675
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ34(29,⋅)\chi_{34} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 34, ( :3), 0.999+0.00333i)(2,\ 34,\ (\ :3),\ 0.999 + 0.00333i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 3.08015−0.00513879i3.08015 - 0.00513879i
L(12)L(\frac12) ≈\approx 3.08015−0.00513879i3.08015 - 0.00513879i
L(4)L(4) 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)
bad2 1+(−2.16−5.22i)T 1 + (-2.16 - 5.22i)T
17 1+(−2.53e3−4.20e3i)T 1 + (-2.53e3 - 4.20e3i)T
good3 1+(−41.5+27.7i)T+(278.−673.i)T2 1 + (-41.5 + 27.7i)T + (278. - 673. i)T^{2}
5 1+(−53.2−10.5i)T+(1.44e4+5.97e3i)T2 1 + (-53.2 - 10.5i)T + (1.44e4 + 5.97e3i)T^{2}
7 1+(−90.3+17.9i)T+(1.08e5−4.50e4i)T2 1 + (-90.3 + 17.9i)T + (1.08e5 - 4.50e4i)T^{2}
11 1+(4.84−7.24i)T+(−6.77e5−1.63e6i)T2 1 + (4.84 - 7.24i)T + (-6.77e5 - 1.63e6i)T^{2}
13 1+(39.4+39.4i)T+4.82e6iT2 1 + (39.4 + 39.4i)T + 4.82e6iT^{2}
19 1+(−3.38e3−8.16e3i)T+(−3.32e7+3.32e7i)T2 1 + (-3.38e3 - 8.16e3i)T + (-3.32e7 + 3.32e7i)T^{2}
23 1+(1.45e4+9.72e3i)T+(5.66e7+1.36e8i)T2 1 + (1.45e4 + 9.72e3i)T + (5.66e7 + 1.36e8i)T^{2}
29 1+(9.04e3−4.54e4i)T+(−5.49e8−2.27e8i)T2 1 + (9.04e3 - 4.54e4i)T + (-5.49e8 - 2.27e8i)T^{2}
31 1+(1.96e4+2.94e4i)T+(−3.39e8+8.19e8i)T2 1 + (1.96e4 + 2.94e4i)T + (-3.39e8 + 8.19e8i)T^{2}
37 1+(−1.84e4+1.23e4i)T+(9.81e8−2.37e9i)T2 1 + (-1.84e4 + 1.23e4i)T + (9.81e8 - 2.37e9i)T^{2}
41 1+(−6.54e3+1.30e3i)T+(4.38e9−1.81e9i)T2 1 + (-6.54e3 + 1.30e3i)T + (4.38e9 - 1.81e9i)T^{2}
43 1+(3.14e4−7.58e4i)T+(−4.46e9−4.46e9i)T2 1 + (3.14e4 - 7.58e4i)T + (-4.46e9 - 4.46e9i)T^{2}
47 1+(1.14e5+1.14e5i)T+1.07e10iT2 1 + (1.14e5 + 1.14e5i)T + 1.07e10iT^{2}
53 1+(−5.46e4−1.32e5i)T+(−1.56e10+1.56e10i)T2 1 + (-5.46e4 - 1.32e5i)T + (-1.56e10 + 1.56e10i)T^{2}
59 1+(−1.91e5−7.94e4i)T+(2.98e10+2.98e10i)T2 1 + (-1.91e5 - 7.94e4i)T + (2.98e10 + 2.98e10i)T^{2}
61 1+(4.58e4+2.30e5i)T+(−4.75e10+1.97e10i)T2 1 + (4.58e4 + 2.30e5i)T + (-4.75e10 + 1.97e10i)T^{2}
67 1−4.19e5iT−9.04e10T2 1 - 4.19e5iT - 9.04e10T^{2}
71 1+(−2.44e5+1.63e5i)T+(4.90e10−1.18e11i)T2 1 + (-2.44e5 + 1.63e5i)T + (4.90e10 - 1.18e11i)T^{2}
73 1+(2.02e4+4.02e3i)T+(1.39e11+5.79e10i)T2 1 + (2.02e4 + 4.02e3i)T + (1.39e11 + 5.79e10i)T^{2}
79 1+(−1.44e5+2.16e5i)T+(−9.30e10−2.24e11i)T2 1 + (-1.44e5 + 2.16e5i)T + (-9.30e10 - 2.24e11i)T^{2}
83 1+(−8.82e5+3.65e5i)T+(2.31e11−2.31e11i)T2 1 + (-8.82e5 + 3.65e5i)T + (2.31e11 - 2.31e11i)T^{2}
89 1+(−3.91e5+3.91e5i)T−4.96e11iT2 1 + (-3.91e5 + 3.91e5i)T - 4.96e11iT^{2}
97 1+(6.46e3−3.25e4i)T+(−7.69e11−3.18e11i)T2 1 + (6.46e3 - 3.25e4i)T + (-7.69e11 - 3.18e11i)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

−14.71340258656295376594416533249, −14.35891877616793225161514745408, −13.22748563774158136616211149735, −12.29464697742746263609391845209, −9.810913656952226111425984777962, −8.415210265026188124858865783590, −7.59375872526357629550620940501, −6.13678386483257893256055337381, −3.61640754624302440469582501833, −1.80586361877009960724271490167, 2.18573476109943796504614507282, 3.59216872540143353564263559295, 5.06330836291640766536752156240, 7.898002679027655052554169423413, 9.330532185179839154917865073833, 9.932494421922539940795887762851, 11.47694896718461114772397312186, 13.40797432035121083036018360535, 13.99750157278227337161805145508, 15.11813393923252800615436449654

Graph of the ZZ-function along the critical line