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Properties

Label 1-259-259.219-r0-0-0
Degree $1$
Conductor $259$
Sign $0.460 - 0.887i$
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  + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + 6-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)10-s + 11-s + (0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)18-s + (−0.939 + 0.342i)19-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + 6-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.5 − 0.866i)10-s + 11-s + (0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)18-s + (−0.939 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(1.20279\)
Root analytic conductor: \(1.20279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{259} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 259,\ (0:\ ),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.022544724 - 1.229383444i\)
\(L(\frac12)\) \(\approx\) \(2.022544724 - 1.229383444i\)
\(L(1)\) \(\approx\) \(1.805214691 - 0.7040833632i\)
\(L(1)\) \(\approx\) \(1.805214691 - 0.7040833632i\)

Euler product

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

−25.62055611205451999102451426928, −25.32799771732546500022090128875, −24.244428485020375816710827407231, −23.47924275808627336723456820167, −22.4313154949880777077120475192, −21.74234142755281274406017610207, −20.68382922511764276216485317890, −19.54407854632877980171417032396, −18.77921248037627165776263726098, −17.54155282922444619032259035667, −16.91623311737316028047718068201, −15.20578852445630876587431435702, −14.75505139592129586912500768839, −14.12635382818446320078500375100, −13.008064491005954333653002783439, −12.25389192753605950020835151012, −11.06400886247919686258573482321, −9.55486269684134923078507884473, −8.42022914594910080381834379951, −7.28667185918871560503813671063, −6.76690801154849399408235651878, −5.66296443971570672731002604772, −4.003718354341678987987495816056, −3.099659307042263435219635202688, −2.03482025285001021206899459735, 1.43738765326373960708460303266, 2.63002021951081202578600208398, 3.93820542293871709664264752503, 4.66709128676889932768371903790, 5.66832646843529668228317926162, 7.252149367915638390309549078516, 8.83193021872486253275553726420, 9.452678684419119896904421094506, 10.38056137607714593403928942937, 11.70618704724010750070888413160, 12.54763910339505400628612389172, 13.56935890496319815917200628202, 14.418059793723003682628444077883, 15.13427585592818263956403583975, 16.34643081150682264927063867372, 17.10126486523941464043880338275, 18.946937714332951212211726302084, 19.57091377512482635756152023500, 20.42708553689517763527402152953, 21.12865499690344248834668464627, 21.81950771745873800613375446251, 22.79730964712336147366526703448, 23.942424523327118801441141655534, 24.92984510846424267935036885434, 25.31377243649401603092729539824

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