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Properties

Label 1-59-59.19-r0-0-0
Degree $1$
Conductor $59$
Sign $-0.581 - 0.813i$
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  + (−0.561 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.581 - 0.813i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ -0.581 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3347362665 - 0.6510329257i\)
\(L(\frac12)\) \(\approx\) \(0.3347362665 - 0.6510329257i\)
\(L(1)\) \(\approx\) \(0.6098631950 - 0.5725676704i\)
\(L(1)\) \(\approx\) \(0.6098631950 - 0.5725676704i\)

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

−33.54557537963119351844581123437, −32.09247580195568366192478057431, −31.52691884900873752009529158775, −29.322965701514118964780052771867, −28.47412855662066857128248272439, −27.21168623573885361930996837783, −26.4960405133830265542701477767, −25.34256772894829297828917381032, −24.56786798613671224079692747908, −22.805823487243888023456377168858, −21.85680845298533600899685117250, −20.76628079240191092560115811503, −19.02709513776293899274921284168, −17.98026689465932588315516181533, −16.85896704501474489757056681188, −15.69104050114255029127429497460, −14.74038103496758241517198975326, −13.71244414377280906397345781852, −11.35370963767665416355532493687, −9.94892459245170246438014012918, −9.27503199953602028505233011490, −7.78489150620068242603026516984, −5.820238542783951596757721572284, −5.15385889612737256864410612523, −2.63975268497039862321130635878, 1.28631158168727281497711970399, 2.62288520083886533045957306416, 4.87094756177036842021155769796, 6.9927140270976323258434810031, 8.11759788477704463445585294297, 9.612525587728435373913973784764, 10.78134090503724918431499965564, 12.40132730123801269582387384774, 13.141810159097590626280264774761, 14.27269842676356926901324510561, 16.85275872447839109168038507596, 17.48650898601926220296349664197, 18.46690241507525938233424849561, 19.87743565525930285995664896744, 20.546132271470260592594328945267, 21.89515400146677343764344738129, 23.33016584521754928667567246624, 24.57010478168608676821971543130, 25.77620663385964236900320124184, 26.65371212945583483974073423663, 28.38480410000025277755047019027, 28.989101175368121772735026046527, 30.00788432183562498723300286868, 30.74648968420538286377143960692, 32.08348728810761424960302791549

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