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Properties

Label 1-59-59.52-r1-0-0
Degree $1$
Conductor $59$
Sign $-0.577 + 0.816i$
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  + (0.370 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2166425081 - 0.4184253402i\)
\(L(\frac12)\) \(\approx\) \(-0.2166425081 - 0.4184253402i\)
\(L(1)\) \(\approx\) \(0.4818734731 - 0.4923280208i\)
\(L(1)\) \(\approx\) \(0.4818734731 - 0.4923280208i\)

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

−33.257626796419570256633349779685, −32.421627345065258890558296057475, −31.18622490622001532210019191223, −29.60093169943891749351881759061, −29.02334572182275404194511683540, −27.24267433242063212528051317154, −26.48383679389417808931784898634, −25.12464085942410437562220164213, −24.23705030187170551055270621401, −22.68888807604500437814004847637, −22.29442940259338328350512761078, −21.356913498129085566175122086389, −18.87149727049699928977138782977, −18.0118330997216399131048477581, −16.813274570916788401096276773033, −15.91610537074739528940503588002, −14.633929538138608299734213053258, −13.27144495520562871638744827077, −12.12588000172841219934242642384, −10.46615055793301507081530368714, −9.14269363565060747018718106497, −7.07279220095090582428329536189, −6.17540475379764428440319769182, −5.15758690611459908356299898312, −3.158608316527950843814506542938, 0.24627041278601672726660186834, 1.98089647941108577453101869444, 4.2972696366607554351373096163, 5.35916828263924095209342322224, 6.85245958572027679313066941645, 9.42098804891719210934385269104, 10.149048044726346741781938021473, 11.6309088861351712513465454599, 12.8157615573663753901389257998, 13.384731365110839176398526934069, 15.36587641665881325709006591558, 17.03718102850737879498118850499, 17.704957737895712717883258862285, 19.308908951235839012320122115, 20.33052022818261517262790897826, 21.66675158851064606673130882244, 22.45596336479753775292758737950, 23.55233622489913863089342286466, 24.56659050099921990559114521668, 26.34873851876043757273644669564, 27.82371742613712322120976142745, 28.68621980528588424532368194261, 29.26769150538593120765719696677, 30.29052895704955771075172503349, 31.76602525320418334690073165802

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