Properties

Degree $2$
Conductor $261$
Sign $-0.800 + 0.599i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0310 − 0.0108i)2-s + (−3.12 + 2.49i)4-s + (−3.79 + 7.87i)5-s + (3.62 − 4.54i)7-s + (−0.140 + 0.222i)8-s + (−0.0321 + 0.285i)10-s + (−3.95 − 6.29i)11-s + (−10.7 − 2.45i)13-s + (0.0632 − 0.180i)14-s + (3.55 − 15.5i)16-s + (4.26 − 4.26i)17-s + (−13.0 − 1.47i)19-s + (−7.77 − 34.0i)20-s + (−0.191 − 0.152i)22-s + (6.32 − 3.04i)23-s + ⋯
L(s)  = 1  + (0.0155 − 0.00543i)2-s + (−0.781 + 0.623i)4-s + (−0.758 + 1.57i)5-s + (0.518 − 0.649i)7-s + (−0.0175 + 0.0278i)8-s + (−0.00321 + 0.0285i)10-s + (−0.359 − 0.572i)11-s + (−0.827 − 0.188i)13-s + (0.00451 − 0.0129i)14-s + (0.222 − 0.974i)16-s + (0.250 − 0.250i)17-s + (−0.687 − 0.0774i)19-s + (−0.388 − 1.70i)20-s + (−0.00869 − 0.00693i)22-s + (0.275 − 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.800 + 0.599i$
Motivic weight: \(2\)
Character: $\chi_{261} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.800 + 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0300664 - 0.0902989i\)
\(L(\frac12)\) \(\approx\) \(0.0300664 - 0.0902989i\)
\(L(2)\) 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 \( 1 + (28.9 - 0.0339i)T \)
good2 \( 1 + (-0.0310 + 0.0108i)T + (3.12 - 2.49i)T^{2} \)
5 \( 1 + (3.79 - 7.87i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + (-3.62 + 4.54i)T + (-10.9 - 47.7i)T^{2} \)
11 \( 1 + (3.95 + 6.29i)T + (-52.4 + 109. i)T^{2} \)
13 \( 1 + (10.7 + 2.45i)T + (152. + 73.3i)T^{2} \)
17 \( 1 + (-4.26 + 4.26i)T - 289iT^{2} \)
19 \( 1 + (13.0 + 1.47i)T + (351. + 80.3i)T^{2} \)
23 \( 1 + (-6.32 + 3.04i)T + (329. - 413. i)T^{2} \)
31 \( 1 + (-2.37 + 0.830i)T + (751. - 599. i)T^{2} \)
37 \( 1 + (17.7 - 28.2i)T + (-593. - 1.23e3i)T^{2} \)
41 \( 1 + (-2.70 - 2.70i)T + 1.68e3iT^{2} \)
43 \( 1 + (11.9 - 34.2i)T + (-1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (7.75 - 4.87i)T + (958. - 1.99e3i)T^{2} \)
53 \( 1 + (39.1 + 18.8i)T + (1.75e3 + 2.19e3i)T^{2} \)
59 \( 1 + 70.7T + 3.48e3T^{2} \)
61 \( 1 + (11.5 + 102. i)T + (-3.62e3 + 828. i)T^{2} \)
67 \( 1 + (-26.2 + 5.98i)T + (4.04e3 - 1.94e3i)T^{2} \)
71 \( 1 + (-93.3 - 21.3i)T + (4.54e3 + 2.18e3i)T^{2} \)
73 \( 1 + (83.5 + 29.2i)T + (4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-51.8 - 32.5i)T + (2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (18.9 + 23.7i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (97.6 - 34.1i)T + (6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (12.7 - 113. i)T + (-9.17e3 - 2.09e3i)T^{2} \)
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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

−12.24553750786091697451574406586, −11.24878253893637843586981050220, −10.62799108323011762684034012173, −9.550521108214151908136509117136, −8.077926618646139885036830776002, −7.64347603007788479411698229499, −6.61837904504711902803193130473, −4.92682875897492542550095742691, −3.77615482978680860149127209720, −2.84510462117472688691537016517, 0.04930339452061811742695886447, 1.72152480395750405860175065186, 4.12731936825491860088002697164, 4.92915717449313671177383190099, 5.61896235030511581085883694104, 7.49693444314905428837715118748, 8.509190619024735081577596516153, 9.070442624001341372512684558015, 10.03203001573802605664251241125, 11.32147559625719384780246376843

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