Properties

Label 1-59-59.34-r1-0-0
Degree $1$
Conductor $59$
Sign $-0.608 - 0.793i$
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  + (−0.267 − 0.963i)2-s + (−0.561 + 0.827i)3-s + (−0.856 + 0.515i)4-s + (0.0541 + 0.998i)5-s + (0.947 + 0.319i)6-s + (−0.161 − 0.986i)7-s + (0.725 + 0.687i)8-s + (−0.370 − 0.928i)9-s + (0.947 − 0.319i)10-s + (−0.468 − 0.883i)11-s + (0.0541 − 0.998i)12-s + (0.370 − 0.928i)13-s + (−0.907 + 0.419i)14-s + (−0.856 − 0.515i)15-s + (0.468 − 0.883i)16-s + (−0.161 + 0.986i)17-s + ⋯
L(s)  = 1  + (−0.267 − 0.963i)2-s + (−0.561 + 0.827i)3-s + (−0.856 + 0.515i)4-s + (0.0541 + 0.998i)5-s + (0.947 + 0.319i)6-s + (−0.161 − 0.986i)7-s + (0.725 + 0.687i)8-s + (−0.370 − 0.928i)9-s + (0.947 − 0.319i)10-s + (−0.468 − 0.883i)11-s + (0.0541 − 0.998i)12-s + (0.370 − 0.928i)13-s + (−0.907 + 0.419i)14-s + (−0.856 − 0.515i)15-s + (0.468 − 0.883i)16-s + (−0.161 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2613219336 - 0.5299732219i\)
\(L(\frac12)\) \(\approx\) \(0.2613219336 - 0.5299732219i\)
\(L(1)\) \(\approx\) \(0.5903208602 - 0.2206955540i\)
\(L(1)\) \(\approx\) \(0.5903208602 - 0.2206955540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.267 - 0.963i)T \)
3 \( 1 + (-0.561 + 0.827i)T \)
5 \( 1 + (0.0541 + 0.998i)T \)
7 \( 1 + (-0.161 - 0.986i)T \)
11 \( 1 + (-0.468 - 0.883i)T \)
13 \( 1 + (0.370 - 0.928i)T \)
17 \( 1 + (-0.161 + 0.986i)T \)
19 \( 1 + (0.647 - 0.762i)T \)
23 \( 1 + (-0.796 - 0.605i)T \)
29 \( 1 + (0.267 - 0.963i)T \)
31 \( 1 + (-0.647 - 0.762i)T \)
37 \( 1 + (0.725 - 0.687i)T \)
41 \( 1 + (0.796 - 0.605i)T \)
43 \( 1 + (-0.468 + 0.883i)T \)
47 \( 1 + (-0.0541 + 0.998i)T \)
53 \( 1 + (-0.947 - 0.319i)T \)
61 \( 1 + (-0.267 - 0.963i)T \)
67 \( 1 + (0.725 + 0.687i)T \)
71 \( 1 + (0.0541 - 0.998i)T \)
73 \( 1 + (-0.907 + 0.419i)T \)
79 \( 1 + (-0.561 - 0.827i)T \)
83 \( 1 + (-0.976 + 0.214i)T \)
89 \( 1 + (-0.267 + 0.963i)T \)
97 \( 1 + (-0.907 - 0.419i)T \)
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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

−33.1924484509649410809633358440, −31.59734101447673947601328315228, −31.111815205810219792064055541131, −28.98384421974651104658109407689, −28.44322264687073399082008468988, −27.44628866043182503259332100634, −25.61332624953963387906244047316, −24.97835657569039127238358151790, −23.969361703572397155968512610976, −23.12080729935926751344002285271, −21.81385302666635412686745013364, −20.005014185326920864247266315344, −18.540901243457091891411891678524, −17.930205958568289981438527793415, −16.54257210146117636493838826490, −15.81355159006930209552858209340, −14.05785395006384797694343933194, −12.8606237402277399519678823860, −11.84040993431735596378692380089, −9.66855054656199130545504026224, −8.487808056947028188125312745684, −7.215388082291607836484712059859, −5.80901356881690239995298027939, −4.86133364381815184977753199852, −1.5972207722135973107399849237, 0.39117932561682263151172017610, 3.0365189133236481193285316408, 4.1136279383198589723410045865, 5.97182021090043240510116540190, 7.93954184417640360967644240935, 9.741573166342397353662487950198, 10.70166669072267028675476300607, 11.22686923946352973352464735907, 13.05045397322703127532278818814, 14.32422772295241871536413970214, 15.91098855221595660880813504992, 17.284115658113225096802829350762, 18.17142583428159956687620405263, 19.57965863903210081742586385986, 20.71142436770378975165382778659, 21.847693167947022854532915302549, 22.61446790574536865190801685874, 23.61663601971367217255798890575, 26.21328498528175696349576615233, 26.47791756275968023443258077244, 27.61358391629641384928339836624, 28.76960021849975266885884291730, 29.7678567876090044438917617358, 30.55551703824662304469760961731, 32.15180727644436874995284515246

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