Properties

Label 1-59-59.48-r0-0-0
Degree $1$
Conductor $59$
Sign $-0.0592 - 0.998i$
Analytic cond. $0.273994$
Root an. cond. $0.273994$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.907 − 0.419i)2-s + (−0.947 − 0.319i)3-s + (0.647 − 0.762i)4-s + (−0.856 − 0.515i)5-s + (−0.994 + 0.108i)6-s + (0.0541 − 0.998i)7-s + (0.267 − 0.963i)8-s + (0.796 + 0.605i)9-s + (−0.994 − 0.108i)10-s + (−0.161 + 0.986i)11-s + (−0.856 + 0.515i)12-s + (0.796 − 0.605i)13-s + (−0.370 − 0.928i)14-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.0541 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.907 − 0.419i)2-s + (−0.947 − 0.319i)3-s + (0.647 − 0.762i)4-s + (−0.856 − 0.515i)5-s + (−0.994 + 0.108i)6-s + (0.0541 − 0.998i)7-s + (0.267 − 0.963i)8-s + (0.796 + 0.605i)9-s + (−0.994 − 0.108i)10-s + (−0.161 + 0.986i)11-s + (−0.856 + 0.515i)12-s + (0.796 − 0.605i)13-s + (−0.370 − 0.928i)14-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.0541 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.0592 - 0.998i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ -0.0592 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6963158353 - 0.7388641126i\)
\(L(\frac12)\) \(\approx\) \(0.6963158353 - 0.7388641126i\)
\(L(1)\) \(\approx\) \(0.9847222608 - 0.5825313506i\)
\(L(1)\) \(\approx\) \(0.9847222608 - 0.5825313506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.907 - 0.419i)T \)
3 \( 1 + (-0.947 - 0.319i)T \)
5 \( 1 + (-0.856 - 0.515i)T \)
7 \( 1 + (0.0541 - 0.998i)T \)
11 \( 1 + (-0.161 + 0.986i)T \)
13 \( 1 + (0.796 - 0.605i)T \)
17 \( 1 + (0.0541 + 0.998i)T \)
19 \( 1 + (-0.725 + 0.687i)T \)
23 \( 1 + (0.976 - 0.214i)T \)
29 \( 1 + (0.907 + 0.419i)T \)
31 \( 1 + (-0.725 - 0.687i)T \)
37 \( 1 + (0.267 + 0.963i)T \)
41 \( 1 + (0.976 + 0.214i)T \)
43 \( 1 + (-0.161 - 0.986i)T \)
47 \( 1 + (-0.856 + 0.515i)T \)
53 \( 1 + (-0.994 + 0.108i)T \)
61 \( 1 + (0.907 - 0.419i)T \)
67 \( 1 + (0.267 - 0.963i)T \)
71 \( 1 + (-0.856 + 0.515i)T \)
73 \( 1 + (-0.370 - 0.928i)T \)
79 \( 1 + (-0.947 + 0.319i)T \)
83 \( 1 + (-0.561 + 0.827i)T \)
89 \( 1 + (0.907 + 0.419i)T \)
97 \( 1 + (-0.370 + 0.928i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.08339255895183652219325613813, −31.87655887409067761201036493731, −31.02949749320264281156133252372, −29.8081168394533759242284408298, −28.68936175528658416179183190347, −27.39283878674455693830349163449, −26.32726557863174877867731284091, −24.831002646204846876214709066338, −23.65525773216460317620129337891, −22.97200339645012886450273529535, −21.81622326485258953155130615968, −21.13366667888380022475295038123, −19.15540190884150284684378288486, −17.923804738065328884379898324529, −16.22520599197829065315150031552, −15.77455777035976110482473061699, −14.53624330584960848759997205596, −12.879547511128034605185635838983, −11.54864044251315541406174232168, −11.11592273639442922446679100479, −8.69781953408936448672651561584, −6.97814737286405356247224140932, −5.8821524289252098117658228648, −4.56264734891741927830888553162, −3.080716466745177044438885084611, 1.30296073650412461488501710141, 3.88970268185709256218637926130, 4.89320685121734620774741356106, 6.46673472598007915799105040477, 7.75889232709729433274191812804, 10.32699574911334330147209455670, 11.15970017771150495980624339259, 12.52785122265456229432356958802, 13.08842413268797087900316811933, 14.9112773157881505504857517797, 16.105562123650978993655732439077, 17.236331775518132215911657557655, 18.89418073219691520888360977230, 20.071846351919314912481118741497, 21.024870635964495413769254116119, 22.650956616173401771216329564054, 23.34655312611416351867237596547, 23.90181042128366831563322993315, 25.34132482476688161955167120188, 27.38797330435639245200100397007, 28.16195576272390118243250815703, 29.220516548853844509865170232185, 30.35272360612160403020815320816, 31.02231090830751403918806459873, 32.636287954096313506963560634970

Graph of the $Z$-function along the critical line