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Properties

Degree $2$
Conductor $261$
Sign $0.182 - 0.983i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 − 0.190i)2-s + (−1.08 + 0.248i)4-s + (−0.141 − 0.113i)5-s + (−1.55 + 6.79i)7-s + (−8.20 + 2.86i)8-s + (−0.260 − 0.163i)10-s + (12.2 + 4.28i)11-s + (9.66 + 20.0i)13-s + (−1.32 + 11.7i)14-s + (−9.27 + 4.46i)16-s + (5.90 + 5.90i)17-s + (−10.7 + 17.0i)19-s + (0.182 + 0.0877i)20-s + (21.5 + 4.90i)22-s + (−15.0 − 18.8i)23-s + ⋯
L(s)  = 1  + (0.843 − 0.0950i)2-s + (−0.271 + 0.0620i)4-s + (−0.0283 − 0.0226i)5-s + (−0.221 + 0.970i)7-s + (−1.02 + 0.358i)8-s + (−0.0260 − 0.0163i)10-s + (1.11 + 0.389i)11-s + (0.743 + 1.54i)13-s + (−0.0946 + 0.840i)14-s + (−0.579 + 0.279i)16-s + (0.347 + 0.347i)17-s + (−0.565 + 0.899i)19-s + (0.00910 + 0.00438i)20-s + (0.977 + 0.223i)22-s + (−0.653 − 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.182 - 0.983i$
Motivic weight: \(2\)
Character: $\chi_{261} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42149 + 1.18210i\)
\(L(\frac12)\) \(\approx\) \(1.42149 + 1.18210i\)
\(L(2)\) 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 \( 1 + (28.5 - 4.93i)T \)
good2 \( 1 + (-1.68 + 0.190i)T + (3.89 - 0.890i)T^{2} \)
5 \( 1 + (0.141 + 0.113i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (1.55 - 6.79i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (-12.2 - 4.28i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (-9.66 - 20.0i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (-5.90 - 5.90i)T + 289iT^{2} \)
19 \( 1 + (10.7 - 17.0i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (15.0 + 18.8i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-35.5 + 4.01i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (23.0 - 8.08i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-14.1 + 14.1i)T - 1.68e3iT^{2} \)
43 \( 1 + (4.31 - 38.2i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (1.53 - 4.38i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-51.0 + 63.9i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + 28.0T + 3.48e3T^{2} \)
61 \( 1 + (3.26 - 2.05i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (2.88 - 5.98i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (-15.5 - 32.3i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-103. - 11.6i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (28.1 + 80.4i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-6.13 - 26.8i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (26.8 - 3.02i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (6.09 + 3.82i)T + (4.08e3 + 8.47e3i)T^{2} \)
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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

−12.09832695058458568710386125684, −11.51606826312507205755652895772, −9.943926631292020832878383947178, −8.997192287597137840093649032134, −8.362993725015938727392372179579, −6.50441248389517300178361870080, −5.95325665619436150132678050468, −4.47228575047909231668193110094, −3.73987375095555040179552002477, −2.06490698991095689970404939993, 0.76966369520124898797270769826, 3.33107143923682147037260413040, 4.01957167947656893906650149445, 5.37445008807065721981445798491, 6.27892154268316878090950332494, 7.44147397178837651230368866217, 8.698353885794394340584993035380, 9.655348330289800101395929074842, 10.69044840287076181585038466238, 11.68707814645364128381264209755

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