Properties

Degree $2$
Conductor $261$
Sign $0.522 + 0.852i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0361 − 0.320i)2-s + (3.79 + 0.866i)4-s + (−1.11 + 0.889i)5-s + (−2.06 − 9.04i)7-s + (0.841 − 2.40i)8-s + (0.245 + 0.390i)10-s + (−3.02 − 8.63i)11-s + (7.25 − 15.0i)13-s + (−2.97 + 0.335i)14-s + (13.2 + 6.40i)16-s + (13.4 + 13.4i)17-s + (−6.13 + 3.85i)19-s + (−5.00 + 2.41i)20-s + (−2.87 + 0.657i)22-s + (26.9 − 33.8i)23-s + ⋯
L(s)  = 1  + (0.0180 − 0.160i)2-s + (0.949 + 0.216i)4-s + (−0.223 + 0.177i)5-s + (−0.294 − 1.29i)7-s + (0.105 − 0.300i)8-s + (0.0245 + 0.0390i)10-s + (−0.274 − 0.785i)11-s + (0.558 − 1.15i)13-s + (−0.212 + 0.0239i)14-s + (0.831 + 0.400i)16-s + (0.793 + 0.793i)17-s + (−0.322 + 0.202i)19-s + (−0.250 + 0.120i)20-s + (−0.130 + 0.0298i)22-s + (1.17 − 1.47i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.522 + 0.852i$
Motivic weight: \(2\)
Character: $\chi_{261} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61780 - 0.905603i\)
\(L(\frac12)\) \(\approx\) \(1.61780 - 0.905603i\)
\(L(2)\) 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 \( 1 + (8.81 - 27.6i)T \)
good2 \( 1 + (-0.0361 + 0.320i)T + (-3.89 - 0.890i)T^{2} \)
5 \( 1 + (1.11 - 0.889i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (2.06 + 9.04i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (3.02 + 8.63i)T + (-94.6 + 75.4i)T^{2} \)
13 \( 1 + (-7.25 + 15.0i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (-13.4 - 13.4i)T + 289iT^{2} \)
19 \( 1 + (6.13 - 3.85i)T + (156. - 325. i)T^{2} \)
23 \( 1 + (-26.9 + 33.8i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (-2.27 + 20.1i)T + (-936. - 213. i)T^{2} \)
37 \( 1 + (-5.54 + 15.8i)T + (-1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-11.0 + 11.0i)T - 1.68e3iT^{2} \)
43 \( 1 + (1.43 - 0.161i)T + (1.80e3 - 411. i)T^{2} \)
47 \( 1 + (52.6 - 18.4i)T + (1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (26.2 + 32.9i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 - 40.0T + 3.48e3T^{2} \)
61 \( 1 + (31.5 - 50.2i)T + (-1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-27.0 - 56.0i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (16.5 - 34.2i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (-10.1 - 90.2i)T + (-5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (68.6 + 24.0i)T + (4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (14.4 - 63.1i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (-3.18 + 28.2i)T + (-7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-59.0 - 93.9i)T + (-4.08e3 + 8.47e3i)T^{2} \)
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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

−11.29034501476742337968559509814, −10.73638329514337696802209425938, −10.15247156870725145712497585653, −8.445893475977800688153954353412, −7.61546244969610407038985269110, −6.71880660365670720332647926770, −5.65239642725905182970214889383, −3.82364060739053956109836541965, −3.02935791791910102263146592186, −1.01092853525088939243614156224, 1.82551315577457884898152239831, 3.06123084585353026329727772349, 4.86988007424861666453028116602, 5.94033633616690053351644180039, 6.87862269097468416889041918596, 7.900876412205949314295966447669, 9.114711169877388852042976983788, 9.887553699915504409855760945078, 11.26811391035148731417840613378, 11.81608383018562437621814432093

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