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Properties

Label 1-59-59.51-r0-0-0
Degree $1$
Conductor $59$
Sign $0.835 - 0.548i$
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  + (−0.947 − 0.319i)2-s + (−0.856 − 0.515i)3-s + (0.796 + 0.605i)4-s + (−0.370 + 0.928i)5-s + (0.647 + 0.762i)6-s + (0.907 − 0.419i)7-s + (−0.561 − 0.827i)8-s + (0.468 + 0.883i)9-s + (0.647 − 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.370 − 0.928i)12-s + (0.468 − 0.883i)13-s + (−0.994 + 0.108i)14-s + (0.796 − 0.605i)15-s + (0.267 + 0.963i)16-s + (0.907 + 0.419i)17-s + ⋯
L(s)  = 1  + (−0.947 − 0.319i)2-s + (−0.856 − 0.515i)3-s + (0.796 + 0.605i)4-s + (−0.370 + 0.928i)5-s + (0.647 + 0.762i)6-s + (0.907 − 0.419i)7-s + (−0.561 − 0.827i)8-s + (0.468 + 0.883i)9-s + (0.647 − 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.370 − 0.928i)12-s + (0.468 − 0.883i)13-s + (−0.994 + 0.108i)14-s + (0.796 − 0.605i)15-s + (0.267 + 0.963i)16-s + (0.907 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.835 - 0.548i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.835 - 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4810410967 - 0.1437985134i\)
\(L(\frac12)\) \(\approx\) \(0.4810410967 - 0.1437985134i\)
\(L(1)\) \(\approx\) \(0.5827605755 - 0.1203877576i\)
\(L(1)\) \(\approx\) \(0.5827605755 - 0.1203877576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.947 - 0.319i)T \)
3 \( 1 + (-0.856 - 0.515i)T \)
5 \( 1 + (-0.370 + 0.928i)T \)
7 \( 1 + (0.907 - 0.419i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (0.468 - 0.883i)T \)
17 \( 1 + (0.907 + 0.419i)T \)
19 \( 1 + (0.976 - 0.214i)T \)
23 \( 1 + (-0.161 + 0.986i)T \)
29 \( 1 + (-0.947 + 0.319i)T \)
31 \( 1 + (0.976 + 0.214i)T \)
37 \( 1 + (-0.561 + 0.827i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (0.267 + 0.963i)T \)
47 \( 1 + (-0.370 - 0.928i)T \)
53 \( 1 + (0.647 + 0.762i)T \)
61 \( 1 + (-0.947 - 0.319i)T \)
67 \( 1 + (-0.561 - 0.827i)T \)
71 \( 1 + (-0.370 - 0.928i)T \)
73 \( 1 + (-0.994 + 0.108i)T \)
79 \( 1 + (-0.856 + 0.515i)T \)
83 \( 1 + (0.0541 + 0.998i)T \)
89 \( 1 + (-0.947 + 0.319i)T \)
97 \( 1 + (-0.994 - 0.108i)T \)
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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

−33.294108196740585695994661467777, −32.02410080610613396822657876442, −30.461689002093972809169984370455, −28.850472088992155354519468217460, −28.14800070144266530333086320620, −27.5538596472241345355851937328, −26.41111006356290135076425767207, −24.85152126969168887461689485950, −24.0235493409905876074953790233, −22.908566151897682290518204204130, −21.03505221739754668382245923813, −20.47802384393614863684915040924, −18.727065103207629702178093838189, −17.66391160129951873074899668929, −16.66851460382872340658882986781, −15.81687435378924429864381827508, −14.605434850187744224338651598890, −12.10844623148261610148130917534, −11.48898282537664414880402342430, −9.90100807435285747758255230035, −8.84180381234602113411721699471, −7.403559896790828203259967344090, −5.709996476243771918569419235612, −4.52663778727195484491999916035, −1.41508389210653202404743129866, 1.2649619380716004988907591948, 3.361155411255152986445977162409, 5.81975128215036787599464041424, 7.29074080507780680541349654216, 8.11945821743760865997701134078, 10.28819653909058379421003197498, 11.1426935838773058168107889517, 11.922636246405650836151457553753, 13.71619846317767177746114549423, 15.4698396170330154686841361570, 16.79594034678165702421816513652, 17.844405247318960849001058129742, 18.608438490174901194688327588180, 19.67620933782080119264969140291, 21.21053509881362190535229742566, 22.38442894462675663319788435579, 23.699422088225342786813394335680, 24.75435074054093054308131333774, 26.182700041301945658781725975287, 27.35274318367125740918032979556, 27.89383780028390029454425229426, 29.509983915402995832575125410895, 30.02367198980436425680083848274, 30.88517603571541791642382804245, 33.1233855198245605461651058574

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