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Properties

Label 2-261-29.12-c0-0-1
Degree $2$
Conductor $261$
Sign $-0.560 + 0.828i$
Analytic cond. $0.130255$
Root an. cond. $0.360909$
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  + (−0.707 − 0.707i)2-s − 1.41i·5-s − 7-s + (−0.707 + 0.707i)8-s + (−1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s − i·13-s + (0.707 + 0.707i)14-s + 1.00·16-s + (−0.707 − 0.707i)17-s − 1.00i·22-s + 1.41·23-s − 1.00·25-s + (−0.707 + 0.707i)26-s + (0.707 + 0.707i)29-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s − 1.41i·5-s − 7-s + (−0.707 + 0.707i)8-s + (−1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s − i·13-s + (0.707 + 0.707i)14-s + 1.00·16-s + (−0.707 − 0.707i)17-s − 1.00i·22-s + 1.41·23-s − 1.00·25-s + (−0.707 + 0.707i)26-s + (0.707 + 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(0.130255\)
Root analytic conductor: \(0.360909\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :0),\ -0.560 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4766005584\)
\(L(\frac12)\) \(\approx\) \(0.4766005584\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - 1.41T + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
97 \( 1 - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.92477545210107226978006814185, −10.80943335051301411452046452952, −9.783793726622232454040017595072, −9.174265410441039523624167500803, −8.531119674978704142199750366646, −7.01589466114184285026331117017, −5.67436059506655455693952650781, −4.61048561012098327644991461910, −2.87185700294022398045455462402, −1.08119077352074327392021995283, 2.88385543173440384690559817456, 3.89058076622138814505087558615, 6.29906451532503022481364672532, 6.54547223258153374591826725715, 7.45113317990104694337439076964, 8.788990231680953979677197181154, 9.423474622233132623928764331999, 10.54290514262918161228838721895, 11.40363112763595231829722397234, 12.50041095762135699720648476133

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