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Properties

Label 2-58-29.28-c1-0-0
Degree 22
Conductor 5858
Sign 0.371−0.928i0.371 - 0.928i
Analytic cond. 0.4631320.463132
Root an. cond. 0.6805380.680538
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s + 5-s − 6-s − 2·7-s − i·8-s + 2·9-s + i·10-s − 5i·11-s − i·12-s − 13-s − 2i·14-s + i·15-s + 16-s − 2i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353i·8-s + 0.666·9-s + 0.316i·10-s − 1.50i·11-s − 0.288i·12-s − 0.277·13-s − 0.534i·14-s + 0.258i·15-s + 0.250·16-s − 0.485i·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.371−0.928i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.371−0.928i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.371−0.928i0.371 - 0.928i
Analytic conductor: 0.4631320.463132
Root analytic conductor: 0.6805380.680538
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ58(57,⋅)\chi_{58} (57, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1/2), 0.371−0.928i)(2,\ 58,\ (\ :1/2),\ 0.371 - 0.928i)

Particular Values

L(1)L(1) ≈\approx 0.705123+0.477391i0.705123 + 0.477391i
L(12)L(\frac12) ≈\approx 0.705123+0.477391i0.705123 + 0.477391i
L(32)L(\frac{3}{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)
bad2 1−iT 1 - iT
29 1+(−5−2i)T 1 + (-5 - 2i)T
good3 1−iT−3T2 1 - iT - 3T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+5iT−11T2 1 + 5iT - 11T^{2}
13 1+T+13T2 1 + T + 13T^{2}
17 1+2iT−17T2 1 + 2iT - 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
31 1−5iT−31T2 1 - 5iT - 31T^{2}
37 1−8iT−37T2 1 - 8iT - 37T^{2}
41 1+10iT−41T2 1 + 10iT - 41T^{2}
43 1+9iT−43T2 1 + 9iT - 43T^{2}
47 1−3iT−47T2 1 - 3iT - 47T^{2}
53 1+T+53T2 1 + T + 53T^{2}
59 1−10T+59T2 1 - 10T + 59T^{2}
61 1+10iT−61T2 1 + 10iT - 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−16iT−73T2 1 - 16iT - 73T^{2}
79 1+iT−79T2 1 + iT - 79T^{2}
83 1−14T+83T2 1 - 14T + 83T^{2}
89 1−14iT−89T2 1 - 14iT - 89T^{2}
97 1+2iT−97T2 1 + 2iT - 97T^{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

−15.87128460803352451050194612739, −14.26119778816102661935931644068, −13.52602251868703231743237573621, −12.19159594137950504415190357871, −10.42310480291414364991838362568, −9.606145269789884566896974019170, −8.304437480803866732936226297063, −6.66752135726560757641321157411, −5.46741702281727274032318267978, −3.67787809784757201819998183332, 2.15684754175300937699526679967, 4.37322746476470901250864601121, 6.35874181585401900380951870048, 7.69276887016559049573974177703, 9.574900076543553179030084851064, 10.11355450344949753919883085106, 11.84645776206177216181121347730, 12.79223661837676310690463501944, 13.41184115269456148166006138874, 14.83758216261447345291267044001

Graph of the ZZ-function along the critical line