Properties

Label 1-59-59.26-r0-0-0
Degree $1$
Conductor $59$
Sign $-0.941 - 0.336i$
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  + (0.267 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.941 - 0.336i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ -0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1268639540 - 0.7313841741i\)
\(L(\frac12)\) \(\approx\) \(0.1268639540 - 0.7313841741i\)
\(L(1)\) \(\approx\) \(0.5262016357 - 0.6940030757i\)
\(L(1)\) \(\approx\) \(0.5262016357 - 0.6940030757i\)

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

−33.21065001692172534675198839195, −32.69268976007025485639271900220, −31.125589249949487247443565041679, −30.11701955382771397888211766624, −28.61621976583427112745448847742, −27.17670730420893192919754639265, −26.47209196610607414657720081265, −25.71032879449648128326695502955, −23.9220826172143939154184930092, −22.98185789854619049635235201439, −22.25182009740668305389573699332, −21.17453930278640736932781357591, −19.3783380313282509507283283589, −17.60534987871695249444049371096, −17.11878543598664321828314757079, −15.693611196920873078775622383483, −14.774316892896561849009190547120, −13.708474012970011286200100965690, −11.88920829997913468163158205588, −10.399872825204851152303542686715, −9.3192582563224842362219281288, −7.23279097754884115001972646616, −6.42600910853829978450001211462, −4.69692492376973427888860669952, −3.60703361425111032379274461775, 1.03075342772306487434805954716, 2.78062737321284252014131427223, 5.01255474335349695246522448965, 5.91502333341394610436113712216, 8.21997507921892488835092203698, 9.408061667175248905526140530, 11.1896598863739785238820069558, 12.23119743546961291273185471456, 12.89783402366847386190366422605, 14.18095394135664463764906663130, 16.12205696497958408256497715136, 17.5308019158322723240717238683, 18.60436384400002922782910213322, 19.58192300175544796661919090765, 20.79042132887935544903414776948, 22.09976091504124777278464775125, 22.93310992722056610439669473084, 24.44539880233825126360818321727, 24.88484854094405842845238539932, 27.24425372783131321089945128245, 28.09006015692452393518548201069, 29.08913267620903821350626723237, 29.685748835851939146070499880063, 31.16006709247143608069883356012, 31.85089739903208850314141786198

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