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Properties

Label 1-261-261.67-r0-0-0
Degree $1$
Conductor $261$
Sign $0.993 - 0.116i$
Analytic cond. $1.21207$
Root an. cond. $1.21207$
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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.988 − 0.149i)5-s + (−0.733 − 0.680i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.0747 + 0.997i)11-s + (0.826 + 0.563i)13-s + (−0.365 + 0.930i)14-s + (0.0747 − 0.997i)16-s − 17-s + (0.222 + 0.974i)19-s + (0.826 − 0.563i)20-s + (0.955 − 0.294i)22-s + (0.365 − 0.930i)23-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.988 − 0.149i)5-s + (−0.733 − 0.680i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.0747 + 0.997i)11-s + (0.826 + 0.563i)13-s + (−0.365 + 0.930i)14-s + (0.0747 − 0.997i)16-s − 17-s + (0.222 + 0.974i)19-s + (0.826 − 0.563i)20-s + (0.955 − 0.294i)22-s + (0.365 − 0.930i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ 0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6233973098 - 0.03634583171i\)
\(L(\frac12)\) \(\approx\) \(0.6233973098 - 0.03634583171i\)
\(L(1)\) \(\approx\) \(0.6331918280 - 0.1835877773i\)
\(L(1)\) \(\approx\) \(0.6331918280 - 0.1835877773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.733 - 0.680i)T \)
11 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 - T \)
19 \( 1 + (0.222 + 0.974i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (0.988 + 0.149i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (0.0747 + 0.997i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + (-0.826 + 0.563i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (-0.955 - 0.294i)T \)
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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

−26.03413208323173761205584197730, −24.92856961341029336147385396899, −24.19039316115388136016003794339, −23.26226824478509025985525378306, −22.53031871045534115387882065765, −21.62353313065518797116492175374, −19.95681700052409820586250993257, −19.27156746856969714544042916532, −18.52275788957218665959251460119, −17.575738799122142399368143702669, −16.27816294321096784480414523925, −15.664718975331126949372508645349, −15.18258014047891270997165406405, −13.71919515608701157630469290192, −12.97187818246694244899069254599, −11.509246024660687336304577394, −10.64087868472573861157111484437, −9.15815901787682199275289363347, −8.57103823637920843147352656551, −7.4843946618330060465521981920, −6.42823843493841875717018490735, −5.532479362856897524768576070442, −4.14408970384104075332082038427, −2.94697029502695031516927692561, −0.61784073591607483073310957305, 1.11518979367311096452239216405, 2.722007866938231350510071075447, 3.98814687614059707873942162309, 4.50425876367448407025428073744, 6.58957669712340265645187231839, 7.649098377612561559973046330502, 8.653045458475864601050750912510, 9.71257584809197600059545110987, 10.67742986085955371197360197995, 11.57363685560545366496761766036, 12.56160756768324320464683357129, 13.26179192548647174341134402444, 14.50109147713723475124325279972, 15.88715305588071883549932180269, 16.590821105607566390991617436572, 17.71129433736499326813516094473, 18.752775761886565113628762916, 19.481965618389365969379622862772, 20.34689801202712015316332205089, 20.85844478030979882112676255505, 22.40108048551314368703790809623, 22.91242419256923269087232609652, 23.68865512821630727628487718693, 25.15485451571930108364675514442, 26.30063567862196273344956518028

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