Properties

Degree $2$
Conductor $261$
Sign $0.449 + 0.893i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.56 − 0.401i)2-s + (8.62 − 1.96i)4-s + (−5.24 − 4.17i)5-s + (2.11 − 9.28i)7-s + (16.3 − 5.73i)8-s + (−20.3 − 12.7i)10-s + (0.700 + 0.245i)11-s + (3.39 + 7.04i)13-s + (3.82 − 33.9i)14-s + (24.1 − 11.6i)16-s + (17.4 + 17.4i)17-s + (−7.96 + 12.6i)19-s + (−53.4 − 25.7i)20-s + (2.59 + 0.592i)22-s + (11.9 + 15.0i)23-s + ⋯
L(s)  = 1  + (1.78 − 0.200i)2-s + (2.15 − 0.492i)4-s + (−1.04 − 0.835i)5-s + (0.302 − 1.32i)7-s + (2.04 − 0.716i)8-s + (−2.03 − 1.27i)10-s + (0.0637 + 0.0222i)11-s + (0.261 + 0.542i)13-s + (0.273 − 2.42i)14-s + (1.51 − 0.727i)16-s + (1.02 + 1.02i)17-s + (−0.419 + 0.667i)19-s + (−2.67 − 1.28i)20-s + (0.117 + 0.0269i)22-s + (0.520 + 0.652i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.449 + 0.893i$
Motivic weight: \(2\)
Character: $\chi_{261} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.449 + 0.893i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.31866 - 2.04501i\)
\(L(\frac12)\) \(\approx\) \(3.31866 - 2.04501i\)
\(L(2)\) 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 \( 1 + (-19.7 + 21.2i)T \)
good2 \( 1 + (-3.56 + 0.401i)T + (3.89 - 0.890i)T^{2} \)
5 \( 1 + (5.24 + 4.17i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (-2.11 + 9.28i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (-0.700 - 0.245i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (-3.39 - 7.04i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (-17.4 - 17.4i)T + 289iT^{2} \)
19 \( 1 + (7.96 - 12.6i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (-11.9 - 15.0i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-0.327 + 0.0369i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (15.2 - 5.34i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (28.0 - 28.0i)T - 1.68e3iT^{2} \)
43 \( 1 + (-0.679 + 6.02i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (1.43 - 4.10i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-37.6 + 47.1i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + 91.1T + 3.48e3T^{2} \)
61 \( 1 + (-6.43 + 4.04i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-29.8 + 62.0i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (-38.1 - 79.2i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (29.1 + 3.28i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-7.28 - 20.8i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (9.19 + 40.2i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (98.0 - 11.0i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-136. - 86.0i)T + (4.08e3 + 8.47e3i)T^{2} \)
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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

−11.88414855194533917422642726030, −11.08814938033413873550100919365, −10.14550770326588272437488614550, −8.315528402604166316006892200488, −7.44546840392832665929579602430, −6.31557168889305147939342337126, −5.01092219499549122151531880555, −4.12332623104456539699051989721, −3.57699832031210228550734526031, −1.38293212217406441462501796679, 2.65433398169023583074583007144, 3.35594853995673386760061938894, 4.73595040475391739925781800333, 5.60993660987770791277924638981, 6.73019117782023036740580297366, 7.58127733465604825473650230312, 8.778232703682792262061373196479, 10.59721766148858697355604621044, 11.45394956850341476686364245357, 12.06371767088153961641741912412

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