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Properties

Label 2-59-59.58-c6-0-6
Degree 22
Conductor 5959
Sign −0.962−0.269i-0.962 - 0.269i
Analytic cond. 13.573113.5731
Root an. cond. 3.684183.68418
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  + 3.85i·2-s + 13.0·3-s + 49.1·4-s − 100.·5-s + 50.4i·6-s − 365.·7-s + 435. i·8-s − 557.·9-s − 386. i·10-s − 61.8i·11-s + 643.·12-s + 4.00e3i·13-s − 1.40e3i·14-s − 1.31e3·15-s + 1.46e3·16-s − 670.·17-s + ⋯
L(s)  = 1  + 0.481i·2-s + 0.485·3-s + 0.768·4-s − 0.803·5-s + 0.233i·6-s − 1.06·7-s + 0.851i·8-s − 0.764·9-s − 0.386i·10-s − 0.0464i·11-s + 0.372·12-s + 1.82i·13-s − 0.512i·14-s − 0.389·15-s + 0.358·16-s − 0.136·17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.962−0.269i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.962−0.269i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.962−0.269i-0.962 - 0.269i
Analytic conductor: 13.573113.5731
Root analytic conductor: 3.684183.68418
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ59(58,⋅)\chi_{59} (58, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 59, ( :3), −0.962−0.269i)(2,\ 59,\ (\ :3),\ -0.962 - 0.269i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.132780+0.967121i0.132780 + 0.967121i
L(12)L(\frac12) ≈\approx 0.132780+0.967121i0.132780 + 0.967121i
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)
bad59 1+(−1.97e5−5.53e4i)T 1 + (-1.97e5 - 5.53e4i)T
good2 1−3.85iT−64T2 1 - 3.85iT - 64T^{2}
3 1−13.0T+729T2 1 - 13.0T + 729T^{2}
5 1+100.T+1.56e4T2 1 + 100.T + 1.56e4T^{2}
7 1+365.T+1.17e5T2 1 + 365.T + 1.17e5T^{2}
11 1+61.8iT−1.77e6T2 1 + 61.8iT - 1.77e6T^{2}
13 1−4.00e3iT−4.82e6T2 1 - 4.00e3iT - 4.82e6T^{2}
17 1+670.T+2.41e7T2 1 + 670.T + 2.41e7T^{2}
19 1+4.25e3T+4.70e7T2 1 + 4.25e3T + 4.70e7T^{2}
23 1−6.23e3iT−1.48e8T2 1 - 6.23e3iT - 1.48e8T^{2}
29 1+741.T+5.94e8T2 1 + 741.T + 5.94e8T^{2}
31 1+4.93e4iT−8.87e8T2 1 + 4.93e4iT - 8.87e8T^{2}
37 1−5.81e4iT−2.56e9T2 1 - 5.81e4iT - 2.56e9T^{2}
41 1−6.74e4T+4.75e9T2 1 - 6.74e4T + 4.75e9T^{2}
43 1+4.06e4iT−6.32e9T2 1 + 4.06e4iT - 6.32e9T^{2}
47 1−1.99e4iT−1.07e10T2 1 - 1.99e4iT - 1.07e10T^{2}
53 1−1.64e5T+2.21e10T2 1 - 1.64e5T + 2.21e10T^{2}
61 1+1.01e5iT−5.15e10T2 1 + 1.01e5iT - 5.15e10T^{2}
67 1−1.20e5iT−9.04e10T2 1 - 1.20e5iT - 9.04e10T^{2}
71 1−4.38e4T+1.28e11T2 1 - 4.38e4T + 1.28e11T^{2}
73 1+4.94e4iT−1.51e11T2 1 + 4.94e4iT - 1.51e11T^{2}
79 1+6.58e5T+2.43e11T2 1 + 6.58e5T + 2.43e11T^{2}
83 1−9.30e5iT−3.26e11T2 1 - 9.30e5iT - 3.26e11T^{2}
89 1−3.83e5iT−4.96e11T2 1 - 3.83e5iT - 4.96e11T^{2}
97 1+3.79e5iT−8.32e11T2 1 + 3.79e5iT - 8.32e11T^{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

−14.61815418270711665977490740149, −13.48524597445712745645536683138, −11.92421758326211646021255157814, −11.25583198196343858395581654613, −9.513694460489204898858328196774, −8.311665609679366652412552271739, −7.09207423306693249364856032490, −6.05766563909247107582226736146, −3.88377926318059913488992404368, −2.38613479711768621972856128922, 0.34625703794940649380845418517, 2.69564030838615853252319213102, 3.56644923458120810266503398676, 5.93577722657179614750357149510, 7.36099560662932445700476211223, 8.547105027619275888114772629513, 10.10038869451754200681223178827, 11.04950775133663448587466865191, 12.30737898534643674901272160082, 13.02618729972761385941363995547

Graph of the ZZ-function along the critical line