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Properties

Label 2-261-261.128-c1-0-16
Degree $2$
Conductor $261$
Sign $0.776 + 0.629i$
Analytic cond. $2.08409$
Root an. cond. $1.44363$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.188 + 0.703i)2-s + (−0.997 − 1.41i)3-s + (1.27 − 0.734i)4-s + (0.0672 + 0.116i)5-s + (0.807 − 0.968i)6-s + (0.205 − 0.355i)7-s + (1.78 + 1.78i)8-s + (−1.00 + 2.82i)9-s + (−0.0692 + 0.0692i)10-s + (−1.30 − 4.86i)11-s + (−2.31 − 1.06i)12-s + (4.99 − 2.88i)13-s + (0.288 + 0.0773i)14-s + (0.0978 − 0.211i)15-s + (0.549 − 0.952i)16-s + (−0.180 + 0.180i)17-s + ⋯
L(s)  = 1  + (0.133 + 0.497i)2-s + (−0.576 − 0.817i)3-s + (0.636 − 0.367i)4-s + (0.0300 + 0.0520i)5-s + (0.329 − 0.395i)6-s + (0.0775 − 0.134i)7-s + (0.631 + 0.631i)8-s + (−0.336 + 0.941i)9-s + (−0.0219 + 0.0219i)10-s + (−0.392 − 1.46i)11-s + (−0.666 − 0.308i)12-s + (1.38 − 0.800i)13-s + (0.0771 + 0.0206i)14-s + (0.0252 − 0.0545i)15-s + (0.137 − 0.238i)16-s + (−0.0438 + 0.0438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ 0.776 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27428 - 0.451808i\)
\(L(\frac12)\) \(\approx\) \(1.27428 - 0.451808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 + 1.41i)T \)
29 \( 1 + (-3.44 - 4.14i)T \)
good2 \( 1 + (-0.188 - 0.703i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.0672 - 0.116i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.205 + 0.355i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.30 + 4.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.99 + 2.88i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.180 - 0.180i)T - 17iT^{2} \)
19 \( 1 + (1.19 + 1.19i)T + 19iT^{2} \)
23 \( 1 + (4.60 - 2.65i)T + (11.5 - 19.9i)T^{2} \)
31 \( 1 + (2.39 - 8.93i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.05 + 3.05i)T - 37iT^{2} \)
41 \( 1 + (2.42 - 9.06i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.99 - 2.14i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.87 - 0.503i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 3.51iT - 53T^{2} \)
59 \( 1 + (6.31 - 3.64i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.76 - 1.81i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.27 + 3.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + (5.84 - 5.84i)T - 73iT^{2} \)
79 \( 1 + (-5.15 + 1.38i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.38 + 1.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.38 - 2.38i)T - 89iT^{2} \)
97 \( 1 + (2.51 + 9.37i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.75361988514414891762941876720, −10.87656969081182033108242488796, −10.52776161863643336557128870586, −8.480683064745441967999440493970, −7.920848365457008953947145965507, −6.60028324508274652030183319293, −6.05137673720024012073478995005, −5.13576099035123339002582656402, −3.03679124020486239550427623055, −1.24621176574483746752027238046, 2.01705422162928773338548869145, 3.70553734810909455025431110855, 4.55171099171047356772599840772, 6.02551015580704432763250800077, 6.96699530236522359681490485800, 8.289959844392586788313043454680, 9.561939019923179359768493813669, 10.35497239128974367022494522289, 11.19714010631959950108517941310, 11.91092777523343224153748388665

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