Properties

Degree $2$
Conductor $261$
Sign $0.999 + 0.00200i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.388 + 3.45i)2-s + (−7.86 + 1.79i)4-s + (−4.44 − 3.54i)5-s + (1.25 − 5.49i)7-s + (−4.66 − 13.3i)8-s + (10.5 − 16.7i)10-s + (6.29 − 17.9i)11-s + (6.68 + 13.8i)13-s + (19.4 + 2.19i)14-s + (15.1 − 7.28i)16-s + (−1.08 + 1.08i)17-s + (−17.6 − 11.1i)19-s + (41.3 + 19.9i)20-s + (64.5 + 14.7i)22-s + (−4.81 − 6.03i)23-s + ⋯
L(s)  = 1  + (0.194 + 1.72i)2-s + (−1.96 + 0.448i)4-s + (−0.889 − 0.709i)5-s + (0.179 − 0.784i)7-s + (−0.583 − 1.66i)8-s + (1.05 − 1.67i)10-s + (0.572 − 1.63i)11-s + (0.514 + 1.06i)13-s + (1.38 + 0.156i)14-s + (0.946 − 0.455i)16-s + (−0.0636 + 0.0636i)17-s + (−0.930 − 0.584i)19-s + (2.06 + 0.995i)20-s + (2.93 + 0.669i)22-s + (−0.209 − 0.262i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00200i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.999 + 0.00200i$
Motivic weight: \(2\)
Character: $\chi_{261} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.999 + 0.00200i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.979455 - 0.000983720i\)
\(L(\frac12)\) \(\approx\) \(0.979455 - 0.000983720i\)
\(L(2)\) 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 \( 1 + (-18.9 + 21.9i)T \)
good2 \( 1 + (-0.388 - 3.45i)T + (-3.89 + 0.890i)T^{2} \)
5 \( 1 + (4.44 + 3.54i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (-1.25 + 5.49i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (-6.29 + 17.9i)T + (-94.6 - 75.4i)T^{2} \)
13 \( 1 + (-6.68 - 13.8i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (1.08 - 1.08i)T - 289iT^{2} \)
19 \( 1 + (17.6 + 11.1i)T + (156. + 325. i)T^{2} \)
23 \( 1 + (4.81 + 6.03i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (5.79 + 51.4i)T + (-936. + 213. i)T^{2} \)
37 \( 1 + (0.580 + 1.65i)T + (-1.07e3 + 853. i)T^{2} \)
41 \( 1 + (0.198 + 0.198i)T + 1.68e3iT^{2} \)
43 \( 1 + (16.5 + 1.86i)T + (1.80e3 + 411. i)T^{2} \)
47 \( 1 + (-2.91 - 1.02i)T + (1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (31.3 - 39.3i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 - 1.15T + 3.48e3T^{2} \)
61 \( 1 + (12.3 + 19.7i)T + (-1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (-2.18 + 4.54i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (12.0 + 24.9i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (8.52 - 75.7i)T + (-5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (-27.2 + 9.54i)T + (4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (-5.13 - 22.4i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (8.12 + 72.1i)T + (-7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (-65.8 + 104. i)T + (-4.08e3 - 8.47e3i)T^{2} \)
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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

−11.79359916012074475404642220010, −10.95943892545005861466463241584, −9.215540609327090537079371892664, −8.483040319904557328590031299042, −7.87631028367382885170219231711, −6.69733669854987350848782386566, −5.95545802851258856102592394957, −4.44188710559907311739460422712, −3.98385175891948523851995500794, −0.49738664928195897046771968995, 1.70870579198596167704256045418, 3.01076722161605189123580398169, 3.98277613267186956459295307686, 5.11831309145381657272251139257, 6.81700586740563004902744471861, 8.158404699513918533891588398699, 9.181179348180615895027421300734, 10.29924776905207479321368791676, 10.81324104800035640099743974667, 11.95598439684649651013042896719

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