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Properties

Label 2-261-29.12-c2-0-23
Degree $2$
Conductor $261$
Sign $-0.425 - 0.904i$
Analytic cond. $7.11173$
Root an. cond. $2.66678$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.909 − 0.909i)2-s − 2.34i·4-s − 4.16i·5-s − 9.68·7-s + (−5.77 + 5.77i)8-s + (−3.78 + 3.78i)10-s + (0.334 + 0.334i)11-s + 12.2i·13-s + (8.81 + 8.81i)14-s + 1.11·16-s + (−6.80 − 6.80i)17-s + (14.6 + 14.6i)19-s − 9.76·20-s − 0.609i·22-s + 10.0·23-s + ⋯
L(s)  = 1  + (−0.454 − 0.454i)2-s − 0.586i·4-s − 0.832i·5-s − 1.38·7-s + (−0.721 + 0.721i)8-s + (−0.378 + 0.378i)10-s + (0.0304 + 0.0304i)11-s + 0.940i·13-s + (0.629 + 0.629i)14-s + 0.0698·16-s + (−0.400 − 0.400i)17-s + (0.771 + 0.771i)19-s − 0.488·20-s − 0.0276i·22-s + 0.438·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(7.11173\)
Root analytic conductor: \(2.66678\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.425 - 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0435979 + 0.0686723i\)
\(L(\frac12)\) \(\approx\) \(0.0435979 + 0.0686723i\)
\(L(2)\) 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 \( 1 + (28.1 - 7.15i)T \)
good2 \( 1 + (0.909 + 0.909i)T + 4iT^{2} \)
5 \( 1 + 4.16iT - 25T^{2} \)
7 \( 1 + 9.68T + 49T^{2} \)
11 \( 1 + (-0.334 - 0.334i)T + 121iT^{2} \)
13 \( 1 - 12.2iT - 169T^{2} \)
17 \( 1 + (6.80 + 6.80i)T + 289iT^{2} \)
19 \( 1 + (-14.6 - 14.6i)T + 361iT^{2} \)
23 \( 1 - 10.0T + 529T^{2} \)
31 \( 1 + (37.3 + 37.3i)T + 961iT^{2} \)
37 \( 1 + (45.0 - 45.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (22.8 - 22.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (17.5 + 17.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (2.20 - 2.20i)T - 2.20e3iT^{2} \)
53 \( 1 + 90.1T + 2.80e3T^{2} \)
59 \( 1 - 90.4T + 3.48e3T^{2} \)
61 \( 1 + (29.4 + 29.4i)T + 3.72e3iT^{2} \)
67 \( 1 + 31.5iT - 4.48e3T^{2} \)
71 \( 1 + 99.8iT - 5.04e3T^{2} \)
73 \( 1 + (3.96 - 3.96i)T - 5.32e3iT^{2} \)
79 \( 1 + (40.3 + 40.3i)T + 6.24e3iT^{2} \)
83 \( 1 + 137.T + 6.88e3T^{2} \)
89 \( 1 + (83.1 + 83.1i)T + 7.92e3iT^{2} \)
97 \( 1 + (79.9 - 79.9i)T - 9.40e3iT^{2} \)
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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

−11.09191756200755102613624881142, −9.846562679565378424984492349740, −9.426054619247798178789213277889, −8.625532280329793384822439697273, −7.04163460003060300376984296267, −6.00697274747154017506562947011, −4.91711141613747991923835909870, −3.34066383667063266520288395341, −1.67011014177566652360565993152, −0.04685008673065721138079459684, 2.92557861774182032329452598611, 3.59267828356608614895019747818, 5.61982907138171082196535467777, 6.87874395260221176241824337663, 7.18824332127601319709501660466, 8.584365998975770390441674698300, 9.418749642238232688572028844251, 10.36897818920995769141282041686, 11.34394819240800989857100417053, 12.72933099684857815743132968999

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