Properties

Label 2-261-29.21-c2-0-9
Degree $2$
Conductor $261$
Sign $0.846 - 0.533i$
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  + (0.144 − 0.412i)2-s + (2.97 + 2.37i)4-s + (1.02 + 2.13i)5-s + (−1.74 − 2.18i)7-s + (2.89 − 1.81i)8-s + (1.02 − 0.115i)10-s + (7.71 + 4.84i)11-s + (8.82 − 2.01i)13-s + (−1.15 + 0.404i)14-s + (3.05 + 13.3i)16-s + (−5.09 + 5.09i)17-s + (2.57 + 22.8i)19-s + (−2.00 + 8.79i)20-s + (3.11 − 2.48i)22-s + (−17.8 − 8.59i)23-s + ⋯
L(s)  = 1  + (0.0722 − 0.206i)2-s + (0.744 + 0.593i)4-s + (0.205 + 0.426i)5-s + (−0.249 − 0.312i)7-s + (0.361 − 0.227i)8-s + (0.102 − 0.0115i)10-s + (0.701 + 0.440i)11-s + (0.679 − 0.155i)13-s + (−0.0825 + 0.0288i)14-s + (0.191 + 0.837i)16-s + (−0.299 + 0.299i)17-s + (0.135 + 1.20i)19-s + (−0.100 + 0.439i)20-s + (0.141 − 0.112i)22-s + (−0.776 − 0.373i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.846 - 0.533i$
Analytic conductor: \(7.11173\)
Root analytic conductor: \(2.66678\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.846 - 0.533i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.99190 + 0.575156i\)
\(L(\frac12)\) \(\approx\) \(1.99190 + 0.575156i\)
\(L(2)\) 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 \( 1 + (-26.8 - 10.9i)T \)
good2 \( 1 + (-0.144 + 0.412i)T + (-3.12 - 2.49i)T^{2} \)
5 \( 1 + (-1.02 - 2.13i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (1.74 + 2.18i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-7.71 - 4.84i)T + (52.4 + 109. i)T^{2} \)
13 \( 1 + (-8.82 + 2.01i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (5.09 - 5.09i)T - 289iT^{2} \)
19 \( 1 + (-2.57 - 22.8i)T + (-351. + 80.3i)T^{2} \)
23 \( 1 + (17.8 + 8.59i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (10.2 - 29.3i)T + (-751. - 599. i)T^{2} \)
37 \( 1 + (-5.78 + 3.63i)T + (593. - 1.23e3i)T^{2} \)
41 \( 1 + (9.53 + 9.53i)T + 1.68e3iT^{2} \)
43 \( 1 + (-47.8 + 16.7i)T + (1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (-44.9 + 71.5i)T + (-958. - 1.99e3i)T^{2} \)
53 \( 1 + (76.8 - 36.9i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 + 54.7T + 3.48e3T^{2} \)
61 \( 1 + (85.8 + 9.67i)T + (3.62e3 + 828. i)T^{2} \)
67 \( 1 + (70.4 + 16.0i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-85.2 + 19.4i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (41.8 + 119. i)T + (-4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (29.8 + 47.4i)T + (-2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (57.5 - 72.2i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (-31.8 + 91.0i)T + (-6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-17.8 + 2.00i)T + (9.17e3 - 2.09e3i)T^{2} \)
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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

−12.07189187049404834793470570207, −10.70463584848531335076280633433, −10.35342881982525135882620043167, −8.902977520112630416800873612375, −7.84291916236330581484633704090, −6.79287278411843065433068476999, −6.09320842229428653564574748325, −4.20839158488503976071341880437, −3.20332859910540920301065357206, −1.73642345723823389981966632297, 1.20162062893095168053128557816, 2.77614745351037598587553482883, 4.49865486337006217141597711232, 5.80531290150465405543982446421, 6.45234582371502668514175983154, 7.62233534905081940993948011523, 8.956641639111785765954592519982, 9.614935051792715004685174378313, 10.96352638920861439754943832676, 11.46914283272984279650845749146

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