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Properties

Label 1-259-259.153-r0-0-0
Degree $1$
Conductor $259$
Sign $0.534 + 0.845i$
Analytic cond. $1.20279$
Root an. cond. $1.20279$
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  + (−0.642 − 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (0.342 + 0.939i)5-s − i·6-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)18-s + (0.642 − 0.766i)19-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + (0.342 + 0.939i)5-s − i·6-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)18-s + (0.642 − 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(1.20279\)
Root analytic conductor: \(1.20279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{259} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 259,\ (0:\ ),\ 0.534 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005520389 + 0.5537944769i\)
\(L(\frac12)\) \(\approx\) \(1.005520389 + 0.5537944769i\)
\(L(1)\) \(\approx\) \(1.001838368 + 0.1944156313i\)
\(L(1)\) \(\approx\) \(1.001838368 + 0.1944156313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.342 + 0.939i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.342 + 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−25.562097716319358784011457251465, −24.85090664296132152615683335983, −24.25229727373420909022120070224, −23.61780565465537589625583370759, −22.1640634510251330379104720244, −20.857973081299558084392599733871, −20.01273041120880440485770067719, −19.18077039438670263067250171626, −18.4492852113689253357704868338, −17.24225229999534457496927458408, −16.684685602612627784766508183177, −15.56855217937976431988974536701, −14.34056493045534322795280843220, −13.89664650612713226021414664426, −12.69960519259573056090042956693, −11.63868006726368759596425036342, −9.77227887945902596401491244555, −9.39130986376780556404982230467, −8.15343080328017437643044120266, −7.67898292499606014775155640679, −6.26210774435628697825606001239, −5.42526605294446619229353411869, −3.85979195065833826657520610818, −2.04301238825798070670193083365, −0.95398403393825529441082315887, 1.893006752523720228047253610797, 2.81340507460452976160830731484, 3.75411354948653250197314291680, 5.040764228073033923271652411098, 7.04283458963907254098725357500, 7.734638265055015585413065635325, 9.120162347147286274226924054153, 9.86023470778898044623241757998, 10.44490858579280859180370017615, 11.62732989203444561359170668307, 12.713772457192914056087755514451, 14.023416397213138092320006875904, 14.6492017229135690842202795372, 15.82008367661300681928888767488, 16.96732734959549970262545178105, 17.889403152076040538678539565031, 18.821148098863753219598287354929, 19.74660298586204364299880824779, 20.3489921881511784991618082359, 21.44601293922230893545178329749, 22.12144768938135222128260224562, 22.77504101480770544224415003087, 24.6591622094354100635299784493, 25.597473985042004452991954254114, 26.12471839485348667721173529893

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