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Properties

Label 2-58-29.28-c1-0-0
Degree $2$
Conductor $58$
Sign $0.371 - 0.928i$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705123 + 0.477391i\)
\(L(\frac12)\) \(\approx\) \(0.705123 + 0.477391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
29 \( 1 + (-5 - 2i)T \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
31 \( 1 - 5iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + 9iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + iT - 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line