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Properties

Label 1-59-59.25-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: $\chi_{59} (25, \cdot )$
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

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

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