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Properties

Label 1-261-261.5-r1-0-0
Degree $1$
Conductor $261$
Sign $-0.636 + 0.771i$
Analytic cond. $28.0483$
Root an. cond. $28.0483$
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  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.955 + 0.294i)5-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + 17-s + (0.900 + 0.433i)19-s + (−0.365 − 0.930i)20-s + (0.826 + 0.563i)22-s + (0.733 − 0.680i)23-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.955 + 0.294i)5-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)8-s + (0.900 + 0.433i)10-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (−0.988 + 0.149i)16-s + 17-s + (0.900 + 0.433i)19-s + (−0.365 − 0.930i)20-s + (0.826 + 0.563i)22-s + (0.733 − 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(28.0483\)
Root analytic conductor: \(28.0483\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (1:\ ),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04058654882 - 0.08606178906i\)
\(L(\frac12)\) \(\approx\) \(-0.04058654882 - 0.08606178906i\)
\(L(1)\) \(\approx\) \(0.5077178653 - 0.2012477500i\)
\(L(1)\) \(\approx\) \(0.5077178653 - 0.2012477500i\)

Euler product

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

−26.29010741079057578918552063229, −25.362001919525833420441404898246, −24.4371240822810632305531197103, −23.66035094050587287514396548203, −23.02618233255108304212636814165, −21.58834990717011493711245605657, −20.58709170826259218504677674406, −19.47477855590667841005250467344, −18.71400861836124890263433916709, −18.14606330533636314531629083358, −16.7324620326778107644791842101, −16.00113926035391503415604908626, −15.3588607264083669039367108106, −14.41615086244517667216314776009, −13.09607259119811012179246367196, −11.781044841054399662043855427279, −11.09712855018018245832079676008, −9.67288225346022266589838924450, −8.79772291346330937718742252049, −7.93706575580517070827003888999, −7.067268578389812660244979283249, −5.64612829286602917086167716077, −4.88829533815224207807739137164, −3.164167498535624683953494562919, −1.52117498504074718703126678564, 0.04446345880377000184735232355, 1.17817559007421859156814988073, 3.01800635102242521326618147284, 3.6719640124334921759408678591, 5.07500138212560800114606203863, 7.0462570354439358213500200545, 7.7218177282903466400335217972, 8.49108804194391508907107432966, 10.146691258262326091370584786743, 10.54378345777307286929395161564, 11.58926077717280025054966376853, 12.58398598814099980953299587943, 13.500271910062152243400888468712, 14.87680223195444555794402555466, 16.03583023149313569787224699912, 16.69743778202309047394448915831, 17.9461038456019381117609319272, 18.6098767781716876667728182360, 19.57058799832401643548842848780, 20.48307207734620646910784332330, 20.893494975190798019610764746586, 22.46722211342163294801089138795, 23.05312504018673886497814758697, 24.046048537270236345151764176811, 25.43542045234663977392825232865

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