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Properties

Label 1-261-261.131-r0-0-0
Degree $1$
Conductor $261$
Sign $-0.252 - 0.967i$
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  + (0.294 − 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

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

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7609945436 - 0.9845509073i\)
\(L(\frac12)\) \(\approx\) \(0.7609945436 - 0.9845509073i\)
\(L(1)\) \(\approx\) \(0.9378573732 - 0.6081157924i\)
\(L(1)\) \(\approx\) \(0.9378573732 - 0.6081157924i\)

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

−25.90086818198910114659531057899, −25.06568792406028702221917379245, −24.32313851905788532606864059326, −23.57418814047132567283500786925, −22.77185129332973138603421931604, −21.65991720550731011606715108195, −20.83772870687859845774794487898, −19.70365045796557217516620350569, −18.53083668711470622626731010310, −17.65619687032335937141333318286, −16.68565497287764743230884343256, −15.909880271432985208213512910123, −14.91135592283173147767526589856, −14.32290035199040539729975182907, −12.89357439400322150235623249038, −12.2308906833123028611171590195, −11.19214563679405438918887275586, −9.49548540480008685868190253364, −8.30236887628294825784433043849, −8.12639514784896432708771642383, −6.54860875092280824231601022740, −5.590901386619624795844723500405, −4.39988175178550000678317737925, −3.70598881912847819685507752153, −1.50228928598313869882579359462, 0.92212965562841012128621564365, 2.40954863927203246179995120089, 3.780698861353993030594951453031, 4.32215593062828364716626172937, 5.85465943806003446718639042449, 7.17062032027423736705898284207, 8.383007152021123529235632677924, 9.4303009700205359541343997290, 10.85640388532416595441473772236, 11.233142755327446247777105385565, 12.05065403760103087719202528291, 13.50820016379089440565607674453, 14.17815230333716912293862483150, 15.01929248515754707432477066694, 16.251605000046670071515571505745, 17.66945305369404404790222326425, 18.35196526747418753321184803678, 19.37356704516523937741073711048, 20.07925362841881933851278252326, 20.97033134182338346962904095401, 21.98345898480979378469205882137, 22.75701474914065402749991377484, 23.633399026350667673661109923584, 24.279320284538239986732575724741, 25.86670259598828635424234562204

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