Properties

Label 1-59-59.7-r0-0-0
Degree $1$
Conductor $59$
Sign $0.884 - 0.466i$
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  + (−0.370 + 0.928i)2-s + (−0.994 − 0.108i)3-s + (−0.725 − 0.687i)4-s + (0.647 − 0.762i)5-s + (0.468 − 0.883i)6-s + (−0.856 − 0.515i)7-s + (0.907 − 0.419i)8-s + (0.976 + 0.214i)9-s + (0.468 + 0.883i)10-s + (0.0541 − 0.998i)11-s + (0.647 + 0.762i)12-s + (0.976 − 0.214i)13-s + (0.796 − 0.605i)14-s + (−0.725 + 0.687i)15-s + (0.0541 + 0.998i)16-s + (−0.856 + 0.515i)17-s + ⋯
L(s)  = 1  + (−0.370 + 0.928i)2-s + (−0.994 − 0.108i)3-s + (−0.725 − 0.687i)4-s + (0.647 − 0.762i)5-s + (0.468 − 0.883i)6-s + (−0.856 − 0.515i)7-s + (0.907 − 0.419i)8-s + (0.976 + 0.214i)9-s + (0.468 + 0.883i)10-s + (0.0541 − 0.998i)11-s + (0.647 + 0.762i)12-s + (0.976 − 0.214i)13-s + (0.796 − 0.605i)14-s + (−0.725 + 0.687i)15-s + (0.0541 + 0.998i)16-s + (−0.856 + 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.884 - 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4978743354 - 0.1232128460i\)
\(L(\frac12)\) \(\approx\) \(0.4978743354 - 0.1232128460i\)
\(L(1)\) \(\approx\) \(0.6292154807 + 0.01358498255i\)
\(L(1)\) \(\approx\) \(0.6292154807 + 0.01358498255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.370 + 0.928i)T \)
3 \( 1 + (-0.994 - 0.108i)T \)
5 \( 1 + (0.647 - 0.762i)T \)
7 \( 1 + (-0.856 - 0.515i)T \)
11 \( 1 + (0.0541 - 0.998i)T \)
13 \( 1 + (0.976 - 0.214i)T \)
17 \( 1 + (-0.856 + 0.515i)T \)
19 \( 1 + (0.267 - 0.963i)T \)
23 \( 1 + (-0.561 - 0.827i)T \)
29 \( 1 + (-0.370 - 0.928i)T \)
31 \( 1 + (0.267 + 0.963i)T \)
37 \( 1 + (0.907 + 0.419i)T \)
41 \( 1 + (-0.561 + 0.827i)T \)
43 \( 1 + (0.0541 + 0.998i)T \)
47 \( 1 + (0.647 + 0.762i)T \)
53 \( 1 + (0.468 - 0.883i)T \)
61 \( 1 + (-0.370 + 0.928i)T \)
67 \( 1 + (0.907 - 0.419i)T \)
71 \( 1 + (0.647 + 0.762i)T \)
73 \( 1 + (0.796 - 0.605i)T \)
79 \( 1 + (-0.994 + 0.108i)T \)
83 \( 1 + (-0.947 + 0.319i)T \)
89 \( 1 + (-0.370 - 0.928i)T \)
97 \( 1 + (0.796 + 0.605i)T \)
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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

−33.01519086716472907180596900143, −31.41886951577264295270699024811, −30.2446186083916687120720766899, −29.24168353068230502842569995296, −28.59307483409685249024462963701, −27.57422583375618220519881071043, −26.253878630956149120426165409227, −25.314632331084026524692971904377, −23.1715538202585891142319989468, −22.41510722130485289957082748264, −21.65667047777952049855189304752, −20.34161718107479655204892672639, −18.642713051365525527076253402461, −18.16910748307692375024129238531, −16.96738995190684659204411387816, −15.57913909620585078953592893992, −13.623336947779849447792802694938, −12.4739193561578331121823484895, −11.32655719257299049834575505785, −10.17400058094007107495507536911, −9.32786274618218436440787053855, −7.06549992544988261202405965075, −5.67863225899467707527916255108, −3.80255226338710792037644315788, −1.993552558425540527868224092198, 0.88983617026657682764085744376, 4.40382969252476865015841139476, 5.89578524825557973753828141448, 6.56412367876226944279211773137, 8.42207853705738286628070569021, 9.74821603493718014808713138140, 10.950632726270927312893068613049, 12.99935293261644459188029780701, 13.619781508236490369493553613686, 15.78121969008632255387516201855, 16.46512713693126997560170746274, 17.384648763754007799330367824874, 18.422799992097695922772714067570, 19.81012208632718608221243109909, 21.637330335666023234309120824957, 22.69717581878790941376380220789, 23.84529186374458858309019503025, 24.57850493694558722651266917420, 25.8993757242072715985572398496, 26.96943696680670727384372303306, 28.450988282597575219733368702173, 28.743929684169540768513809832152, 30.20955061461945335439454917822, 32.30974257468215292355411162934, 32.685063793923200054186868656345

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