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Properties

Label 1-261-261.101-r0-0-0
Degree $1$
Conductor $261$
Sign $0.998 - 0.0612i$
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  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.955 + 0.294i)5-s + (0.0747 + 0.997i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.149 − 0.988i)11-s + (−0.365 − 0.930i)13-s + (0.680 − 0.733i)14-s + (−0.988 − 0.149i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.365 + 0.930i)20-s + (−0.826 + 0.563i)22-s + (0.733 + 0.680i)23-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)2-s + (−0.0747 + 0.997i)4-s + (0.955 + 0.294i)5-s + (0.0747 + 0.997i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.149 − 0.988i)11-s + (−0.365 − 0.930i)13-s + (0.680 − 0.733i)14-s + (−0.988 − 0.149i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.365 + 0.930i)20-s + (−0.826 + 0.563i)22-s + (0.733 + 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.998 - 0.0612i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ 0.998 - 0.0612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026854405 - 0.03148635867i\)
\(L(\frac12)\) \(\approx\) \(1.026854405 - 0.03148635867i\)
\(L(1)\) \(\approx\) \(0.9086016279 - 0.1075338805i\)
\(L(1)\) \(\approx\) \(0.9086016279 - 0.1075338805i\)

Euler product

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

−26.03951933216529887821024899008, −24.93921025152976517120622050592, −24.38168091434235293812772439070, −23.28874712054256267271452149416, −22.51583107671231621974771203866, −21.11589030309630017348697134698, −20.2562653309536004730269159648, −19.421600813854596456405539069935, −18.11570005383515819808828572625, −17.517765277936022821091172257074, −16.74532525406839025222317060285, −15.928221055714633404816701787194, −14.52477141782753408172862087289, −13.994456799648247811108527901459, −12.95916909150887159901733414648, −11.41448075853668886245181291871, −10.22691138986506024749221832047, −9.56340728203335518597071945338, −8.69336047755218775462063783334, −7.03273136651148854693211488380, −6.91239667052654854334565281944, −5.22833961055318779083151103377, −4.51027817970663472208437772827, −2.30520831410998909338322082489, −1.05416794485816519865711021291, 1.38864481190021920141353509037, 2.56768514471626678155251688496, 3.4740636233616614679921975272, 5.339466082518265282825337851941, 6.24872126722584308079633220632, 7.80753278902902957512904259738, 8.72859235962658978119019776350, 9.657835088529193682761935809904, 10.52934497622839980723396016556, 11.49822922014225346456274281771, 12.57788724971532707717635029863, 13.38235465515699810331810317825, 14.58216341041001892360664708560, 15.74689235244749513253191437049, 17.01007081309435149939399911756, 17.60157294750982988293866428477, 18.68024246207727881371759682177, 19.12087948991137441890786033627, 20.44960700798432080378142554299, 21.33766945539603511392862735514, 21.92187227564303194794192255179, 22.67705812152283679841054645113, 24.443063905249806056621848878958, 25.1384957773990283606481264871, 25.90299484966807551710646607137

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