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Properties

Label 1-59-59.40-r1-0-0
Degree $1$
Conductor $59$
Sign $0.970 - 0.240i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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  + (0.561 + 0.827i)2-s + (0.0541 − 0.998i)3-s + (−0.370 + 0.928i)4-s + (0.907 − 0.419i)5-s + (0.856 − 0.515i)6-s + (0.267 − 0.963i)7-s + (−0.976 + 0.214i)8-s + (−0.994 − 0.108i)9-s + (0.856 + 0.515i)10-s + (0.725 − 0.687i)11-s + (0.907 + 0.419i)12-s + (0.994 − 0.108i)13-s + (0.947 − 0.319i)14-s + (−0.370 − 0.928i)15-s + (−0.725 − 0.687i)16-s + (0.267 + 0.963i)17-s + ⋯
L(s)  = 1  + (0.561 + 0.827i)2-s + (0.0541 − 0.998i)3-s + (−0.370 + 0.928i)4-s + (0.907 − 0.419i)5-s + (0.856 − 0.515i)6-s + (0.267 − 0.963i)7-s + (−0.976 + 0.214i)8-s + (−0.994 − 0.108i)9-s + (0.856 + 0.515i)10-s + (0.725 − 0.687i)11-s + (0.907 + 0.419i)12-s + (0.994 − 0.108i)13-s + (0.947 − 0.319i)14-s + (−0.370 − 0.928i)15-s + (−0.725 − 0.687i)16-s + (0.267 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.345298999 - 0.2868330070i\)
\(L(\frac12)\) \(\approx\) \(2.345298999 - 0.2868330070i\)
\(L(1)\) \(\approx\) \(1.635834736 + 0.0007042275750i\)
\(L(1)\) \(\approx\) \(1.635834736 + 0.0007042275750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.561 + 0.827i)T \)
3 \( 1 + (0.0541 - 0.998i)T \)
5 \( 1 + (0.907 - 0.419i)T \)
7 \( 1 + (0.267 - 0.963i)T \)
11 \( 1 + (0.725 - 0.687i)T \)
13 \( 1 + (0.994 - 0.108i)T \)
17 \( 1 + (0.267 + 0.963i)T \)
19 \( 1 + (0.796 - 0.605i)T \)
23 \( 1 + (-0.468 + 0.883i)T \)
29 \( 1 + (-0.561 + 0.827i)T \)
31 \( 1 + (-0.796 - 0.605i)T \)
37 \( 1 + (-0.976 - 0.214i)T \)
41 \( 1 + (0.468 + 0.883i)T \)
43 \( 1 + (0.725 + 0.687i)T \)
47 \( 1 + (-0.907 - 0.419i)T \)
53 \( 1 + (-0.856 + 0.515i)T \)
61 \( 1 + (0.561 + 0.827i)T \)
67 \( 1 + (-0.976 + 0.214i)T \)
71 \( 1 + (0.907 + 0.419i)T \)
73 \( 1 + (0.947 - 0.319i)T \)
79 \( 1 + (0.0541 + 0.998i)T \)
83 \( 1 + (0.161 + 0.986i)T \)
89 \( 1 + (0.561 - 0.827i)T \)
97 \( 1 + (0.947 + 0.319i)T \)
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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

−32.54706016972592088085905733151, −31.30274907090384950596090011535, −30.44915783423508845808788891143, −29.02359323411704478689338788130, −28.19030362848734836577219132995, −27.26722119477692103284771005862, −25.76202014317208737285984987203, −24.69537743491032665421472814482, −22.69465585272214300215906843261, −22.28603316857540412802559041746, −21.049477280961861832128305210554, −20.50340101361235764822132248368, −18.76853900364545850083008740444, −17.74085213113234934124371936516, −15.922798638683542377648443113850, −14.65012266833741275872079815550, −13.94311616392036592484868624325, −12.177705877095882140893911140548, −11.043916778277531720034601868889, −9.80094458205197245782799392119, −9.00405234113446009985351429128, −6.10834805224506871238937647489, −5.06533678106952528112782918515, −3.467105205884490341900673345225, −2.02384933004330826616161733294, 1.29170792827496470509937896809, 3.59089459485363190620406519049, 5.545043255743529194368978009370, 6.529774318551345460880463915, 7.862688149911566075411916717955, 9.07207122190875116189218344685, 11.2867805178562792073703011428, 12.87742720919424605735765338816, 13.64740540856671433828092665228, 14.40766843819712199437135599304, 16.39509477570367354529796780069, 17.30153029155836501569256418453, 18.15090363810027495402469757615, 19.91042372375411723887017552250, 21.17059323398520317649940935675, 22.461788674606573308013911279109, 23.796062974244998345619202562700, 24.30518952925992075774340689369, 25.508473486276053159169792769255, 26.24942730144479081508082089577, 27.969209572706900139054554397784, 29.6382831519249237568468643581, 30.097075898057914574709227264638, 31.317367700093015845033846921306, 32.721647858169174083342396541545

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