Properties

Degree $2$
Conductor $261$
Sign $0.442 - 0.896i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.58 − 0.290i)2-s + (2.68 + 0.611i)4-s + (2.49 − 1.98i)5-s + (1.30 + 5.70i)7-s + (3.06 + 1.07i)8-s + (−7.00 + 4.40i)10-s + (−16.1 + 5.63i)11-s + (2.84 − 5.90i)13-s + (−1.70 − 15.1i)14-s + (−17.5 − 8.43i)16-s + (14.7 − 14.7i)17-s + (9.65 + 15.3i)19-s + (7.89 − 3.80i)20-s + (43.2 − 9.86i)22-s + (−18.5 + 23.2i)23-s + ⋯
L(s)  = 1  + (−1.29 − 0.145i)2-s + (0.670 + 0.152i)4-s + (0.498 − 0.397i)5-s + (0.185 + 0.814i)7-s + (0.383 + 0.134i)8-s + (−0.700 + 0.440i)10-s + (−1.46 + 0.512i)11-s + (0.218 − 0.453i)13-s + (−0.121 − 1.07i)14-s + (−1.09 − 0.527i)16-s + (0.869 − 0.869i)17-s + (0.508 + 0.808i)19-s + (0.394 − 0.190i)20-s + (1.96 − 0.448i)22-s + (−0.807 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.442 - 0.896i$
Motivic weight: \(2\)
Character: $\chi_{261} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.442 - 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.585422 + 0.363793i\)
\(L(\frac12)\) \(\approx\) \(0.585422 + 0.363793i\)
\(L(2)\) 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 \( 1 + (-19.9 - 21.0i)T \)
good2 \( 1 + (2.58 + 0.290i)T + (3.89 + 0.890i)T^{2} \)
5 \( 1 + (-2.49 + 1.98i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (-1.30 - 5.70i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (16.1 - 5.63i)T + (94.6 - 75.4i)T^{2} \)
13 \( 1 + (-2.84 + 5.90i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (-14.7 + 14.7i)T - 289iT^{2} \)
19 \( 1 + (-9.65 - 15.3i)T + (-156. + 325. i)T^{2} \)
23 \( 1 + (18.5 - 23.2i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (13.3 + 1.50i)T + (936. + 213. i)T^{2} \)
37 \( 1 + (-17.2 - 6.03i)T + (1.07e3 + 853. i)T^{2} \)
41 \( 1 + (-44.4 - 44.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-7.00 - 62.1i)T + (-1.80e3 + 411. i)T^{2} \)
47 \( 1 + (-16.6 - 47.6i)T + (-1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (12.0 + 15.1i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 + 47.9T + 3.48e3T^{2} \)
61 \( 1 + (-41.0 - 25.8i)T + (1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (6.15 + 12.7i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (-11.6 + 24.1i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (12.2 - 1.38i)T + (5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (-33.5 + 95.7i)T + (-4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (-9.73 + 42.6i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (53.9 + 6.07i)T + (7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (124. - 78.0i)T + (4.08e3 - 8.47e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.72933811928180169622629088699, −10.66700918136437331690530005210, −9.759801150448300259704062551628, −9.291767065710109578055883302010, −7.981867602945328684215567220493, −7.64895068522450384186460540544, −5.75743347315364374261466017188, −4.98417825875718605028287451766, −2.76562330065622360351941912973, −1.38132518525423599956152617143, 0.58774311999463166153897772464, 2.33949297787309776873513226657, 4.16298748615700386837201698677, 5.72166859423342694422446194659, 6.95954790434476186513668571415, 7.86235418977798557486692325379, 8.580498670710614915045140551191, 9.834669985296160542911454219394, 10.46255517588239026300899846936, 10.97542919588777607901598347029

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