Properties

Degree $2$
Conductor $261$
Sign $0.0607 - 0.998i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.20 − 0.770i)2-s + (1.12 − 0.897i)4-s + (−2.64 + 5.48i)5-s + (−3.22 + 4.04i)7-s + (−3.17 + 5.05i)8-s + (−1.59 + 14.1i)10-s + (−3.16 − 5.03i)11-s + (14.6 + 3.35i)13-s + (−3.98 + 11.4i)14-s + (−4.38 + 19.1i)16-s + (−22.0 + 22.0i)17-s + (−0.835 − 0.0941i)19-s + (1.95 + 8.54i)20-s + (−10.8 − 8.65i)22-s + (21.1 − 10.1i)23-s + ⋯
L(s)  = 1  + (1.10 − 0.385i)2-s + (0.281 − 0.224i)4-s + (−0.528 + 1.09i)5-s + (−0.461 + 0.578i)7-s + (−0.397 + 0.632i)8-s + (−0.159 + 1.41i)10-s + (−0.287 − 0.457i)11-s + (1.12 + 0.257i)13-s + (−0.284 + 0.814i)14-s + (−0.273 + 1.19i)16-s + (−1.29 + 1.29i)17-s + (−0.0439 − 0.00495i)19-s + (0.0975 + 0.427i)20-s + (−0.493 − 0.393i)22-s + (0.919 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0607 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0607 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.0607 - 0.998i$
Motivic weight: \(2\)
Character: $\chi_{261} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.0607 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39087 + 1.30883i\)
\(L(\frac12)\) \(\approx\) \(1.39087 + 1.30883i\)
\(L(2)\) 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 \( 1 + (-18.7 + 22.1i)T \)
good2 \( 1 + (-2.20 + 0.770i)T + (3.12 - 2.49i)T^{2} \)
5 \( 1 + (2.64 - 5.48i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + (3.22 - 4.04i)T + (-10.9 - 47.7i)T^{2} \)
11 \( 1 + (3.16 + 5.03i)T + (-52.4 + 109. i)T^{2} \)
13 \( 1 + (-14.6 - 3.35i)T + (152. + 73.3i)T^{2} \)
17 \( 1 + (22.0 - 22.0i)T - 289iT^{2} \)
19 \( 1 + (0.835 + 0.0941i)T + (351. + 80.3i)T^{2} \)
23 \( 1 + (-21.1 + 10.1i)T + (329. - 413. i)T^{2} \)
31 \( 1 + (-21.3 + 7.47i)T + (751. - 599. i)T^{2} \)
37 \( 1 + (-4.25 + 6.77i)T + (-593. - 1.23e3i)T^{2} \)
41 \( 1 + (-8.24 - 8.24i)T + 1.68e3iT^{2} \)
43 \( 1 + (3.09 - 8.85i)T + (-1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (-22.1 + 13.8i)T + (958. - 1.99e3i)T^{2} \)
53 \( 1 + (47.7 + 22.9i)T + (1.75e3 + 2.19e3i)T^{2} \)
59 \( 1 + 17.4T + 3.48e3T^{2} \)
61 \( 1 + (-6.87 - 60.9i)T + (-3.62e3 + 828. i)T^{2} \)
67 \( 1 + (26.2 - 5.98i)T + (4.04e3 - 1.94e3i)T^{2} \)
71 \( 1 + (-4.93 - 1.12i)T + (4.54e3 + 2.18e3i)T^{2} \)
73 \( 1 + (53.0 + 18.5i)T + (4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-74.2 - 46.6i)T + (2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (-48.3 - 60.6i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (-138. + 48.3i)T + (6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-0.115 + 1.02i)T + (-9.17e3 - 2.09e3i)T^{2} \)
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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

−12.01694839509067864093597348447, −11.11639394394391392991391966459, −10.66529441382620543313878491511, −8.977678918353395559903787516045, −8.174479657113214867879412969816, −6.55801705474962415073841589306, −5.99297514560576612429823144662, −4.42440877987986421739392813062, −3.44582698772107217525716951309, −2.51033075782731305502412983710, 0.70203443178470764684799315204, 3.25391736051372654615735547280, 4.45529770961560441439556335508, 5.02723889530300806738900196238, 6.38345144848968183570408206668, 7.29558276175557895855048905121, 8.655139041783949795490821625936, 9.442825468010156269963965475629, 10.75731411895206577758737668919, 11.88340347802809902469614640711

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