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Properties

Label 1-261-261.205-r1-0-0
Degree $1$
Conductor $261$
Sign $0.967 + 0.252i$
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  + (−0.294 − 0.955i)2-s + (−0.826 + 0.563i)4-s + (0.733 + 0.680i)5-s + (0.826 + 0.563i)7-s + (0.781 + 0.623i)8-s + (0.433 − 0.900i)10-s + (−0.930 − 0.365i)11-s + (0.988 − 0.149i)13-s + (0.294 − 0.955i)14-s + (0.365 − 0.930i)16-s − i·17-s + (−0.433 + 0.900i)19-s + (−0.988 − 0.149i)20-s + (−0.0747 + 0.997i)22-s + (0.955 + 0.294i)23-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)2-s + (−0.826 + 0.563i)4-s + (0.733 + 0.680i)5-s + (0.826 + 0.563i)7-s + (0.781 + 0.623i)8-s + (0.433 − 0.900i)10-s + (−0.930 − 0.365i)11-s + (0.988 − 0.149i)13-s + (0.294 − 0.955i)14-s + (0.365 − 0.930i)16-s − i·17-s + (−0.433 + 0.900i)19-s + (−0.988 − 0.149i)20-s + (−0.0747 + 0.997i)22-s + (0.955 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(28.0483\)
Root analytic conductor: \(28.0483\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (1:\ ),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.773526488 + 0.2271399651i\)
\(L(\frac12)\) \(\approx\) \(1.773526488 + 0.2271399651i\)
\(L(1)\) \(\approx\) \(1.080288006 - 0.1525186045i\)
\(L(1)\) \(\approx\) \(1.080288006 - 0.1525186045i\)

Euler product

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

−25.68255586952537630763072192394, −24.73197833981629840720737776601, −23.67008304770134627130831427842, −23.52374674850214272283201877348, −21.95534736686305943733344389442, −21.0223229059375798624462011942, −20.206911176120728607308280761472, −18.84961487443123646406851105002, −17.89250362467853975979957226382, −17.27470816121511037218002378430, −16.46728730492152106433053907837, −15.388373539479383295536707411909, −14.49882896235833803101683392671, −13.35379398906071662585589054681, −12.95859414524266771429158603578, −11.01436208986038910694956319700, −10.21990610789806729909820546606, −8.96654189487507547457964831266, −8.285880637321972847687629511668, −7.19110667815339595839265178733, −5.988751050050016034222175201781, −5.05662990031384506289097932605, −4.155674999986242571835011347956, −1.916072467802732299743888554158, −0.70202687104289180477971771045, 1.25607900737664128225862694076, 2.40812369523245342766512063218, 3.316032262358198438963825659965, 4.91707689959419309536422667275, 5.86514652510624684698239976428, 7.516491547124941453424114656, 8.54297482081461367473192740795, 9.48741625625765802896958758682, 10.727473398847183063665418076021, 11.08658993055711277601221612004, 12.35169248225683099364184027483, 13.44913346232779600500096642569, 14.13188769166925775728196891884, 15.30298179106688871187283988776, 16.65867250694591815507084350457, 17.81621609363324706624633545515, 18.41081471615431772785284350118, 18.9059143849563886187404570495, 20.50361386932482052344721237066, 21.06713196356198760605929797048, 21.71702277422165719037957416581, 22.76447739686741051722812245070, 23.600613847652650992519502137613, 25.130994133140660204148696094321, 25.70131655700610897265664999673

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