Properties

Degree $2$
Conductor $261$
Sign $0.119 + 0.992i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.32 − 0.814i)2-s + (1.62 − 1.29i)4-s + (3.83 − 7.95i)5-s + (2.23 − 2.80i)7-s + (−2.52 + 4.01i)8-s + (2.43 − 21.6i)10-s + (−4.04 − 6.43i)11-s + (−7.19 − 1.64i)13-s + (2.92 − 8.35i)14-s + (−4.44 + 19.4i)16-s + (−1.60 + 1.60i)17-s + (33.1 + 3.72i)19-s + (−4.08 − 17.8i)20-s + (−14.6 − 11.6i)22-s + (20.4 − 9.84i)23-s + ⋯
L(s)  = 1  + (1.16 − 0.407i)2-s + (0.405 − 0.323i)4-s + (0.766 − 1.59i)5-s + (0.319 − 0.400i)7-s + (−0.315 + 0.501i)8-s + (0.243 − 2.16i)10-s + (−0.367 − 0.585i)11-s + (−0.553 − 0.126i)13-s + (0.208 − 0.596i)14-s + (−0.278 + 1.21i)16-s + (−0.0945 + 0.0945i)17-s + (1.74 + 0.196i)19-s + (−0.204 − 0.893i)20-s + (−0.665 − 0.531i)22-s + (0.888 − 0.427i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.119 + 0.992i$
Motivic weight: \(2\)
Character: $\chi_{261} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.119 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.25791 - 2.00291i\)
\(L(\frac12)\) \(\approx\) \(2.25791 - 2.00291i\)
\(L(2)\) 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 \( 1 + (14.4 - 25.1i)T \)
good2 \( 1 + (-2.32 + 0.814i)T + (3.12 - 2.49i)T^{2} \)
5 \( 1 + (-3.83 + 7.95i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + (-2.23 + 2.80i)T + (-10.9 - 47.7i)T^{2} \)
11 \( 1 + (4.04 + 6.43i)T + (-52.4 + 109. i)T^{2} \)
13 \( 1 + (7.19 + 1.64i)T + (152. + 73.3i)T^{2} \)
17 \( 1 + (1.60 - 1.60i)T - 289iT^{2} \)
19 \( 1 + (-33.1 - 3.72i)T + (351. + 80.3i)T^{2} \)
23 \( 1 + (-20.4 + 9.84i)T + (329. - 413. i)T^{2} \)
31 \( 1 + (-5.30 + 1.85i)T + (751. - 599. i)T^{2} \)
37 \( 1 + (-8.64 + 13.7i)T + (-593. - 1.23e3i)T^{2} \)
41 \( 1 + (-42.1 - 42.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (-1.74 + 4.98i)T + (-1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (60.4 - 37.9i)T + (958. - 1.99e3i)T^{2} \)
53 \( 1 + (-78.2 - 37.7i)T + (1.75e3 + 2.19e3i)T^{2} \)
59 \( 1 + 67.9T + 3.48e3T^{2} \)
61 \( 1 + (-5.72 - 50.8i)T + (-3.62e3 + 828. i)T^{2} \)
67 \( 1 + (29.4 - 6.72i)T + (4.04e3 - 1.94e3i)T^{2} \)
71 \( 1 + (17.3 + 3.95i)T + (4.54e3 + 2.18e3i)T^{2} \)
73 \( 1 + (-29.8 - 10.4i)T + (4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (38.0 + 23.9i)T + (2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (3.87 + 4.85i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (-19.6 + 6.88i)T + (6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-5.86 + 52.0i)T + (-9.17e3 - 2.09e3i)T^{2} \)
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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

−11.88218915236758587836807878248, −10.87073777953813723583950201508, −9.566243250760383965854590257455, −8.766031021254344122876748889671, −7.67360747024705867524286751104, −5.86103836831198743365029895274, −5.16344416360536154753591963684, −4.42162018218182133277693657963, −2.90646353595429106680112498787, −1.22926316737210413351719267894, 2.39840575128916110904041155058, 3.41637311430209695038123227876, 5.00788877871172990777458765655, 5.76262623394966130921470081341, 6.87031393619153578273957180340, 7.47726486099672716062137574826, 9.432107444504262825059281256735, 10.01365867062927244284039602004, 11.24195529901486799728751400852, 12.04681618879209781722338298721

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