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Properties

Label 2-260-65.4-c1-0-2
Degree $2$
Conductor $260$
Sign $0.956 + 0.292i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
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  + (−0.180 − 0.104i)3-s + (1.60 − 1.56i)5-s + (1.72 + 2.99i)7-s + (−1.47 − 2.56i)9-s + (0.625 + 0.360i)11-s + (3.18 − 1.69i)13-s + (−0.452 + 0.115i)15-s + (−3.10 + 1.79i)17-s + (6.51 − 3.76i)19-s − 0.721i·21-s + (2.05 + 1.18i)23-s + (0.125 − 4.99i)25-s + 1.24i·27-s + (−3.68 + 6.37i)29-s − 0.668i·31-s + ⋯
L(s)  = 1  + (−0.104 − 0.0602i)3-s + (0.715 − 0.698i)5-s + (0.653 + 1.13i)7-s + (−0.492 − 0.853i)9-s + (0.188 + 0.108i)11-s + (0.883 − 0.468i)13-s + (−0.116 + 0.0297i)15-s + (−0.754 + 0.435i)17-s + (1.49 − 0.862i)19-s − 0.157i·21-s + (0.428 + 0.247i)23-s + (0.0250 − 0.999i)25-s + 0.239i·27-s + (−0.683 + 1.18i)29-s − 0.120i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41441 - 0.211621i\)
\(L(\frac12)\) \(\approx\) \(1.41441 - 0.211621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.60 + 1.56i)T \)
13 \( 1 + (-3.18 + 1.69i)T \)
good3 \( 1 + (0.180 + 0.104i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.72 - 2.99i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.625 - 0.360i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.10 - 1.79i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.51 + 3.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.05 - 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.68 - 6.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.668iT - 31T^{2} \)
37 \( 1 + (3.36 - 5.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.32 + 3.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.06 - 4.07i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.40T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + (4.20 - 2.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.68 - 4.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.80 - 13.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.10 - 2.37i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.36T + 73T^{2} \)
79 \( 1 - 5.20T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + (-4.20 - 2.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.83 + 3.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.84051520525299707894034333150, −11.28673979965643109570922009825, −9.848422234464662102394945589102, −8.847702490945227986803086233370, −8.519502785247397476504067147677, −6.78859440677974890382921001345, −5.68635478371143725221821028120, −5.02328468854763650252714725260, −3.19164129382976587013653879256, −1.51819065312308178021858497700, 1.75683604869581378162560476597, 3.43870905634417617010528586958, 4.84011212626120241442203237095, 5.99424160715074085990629934944, 7.11646174144086377231267634257, 7.999217340061549933878430745229, 9.273710149678043273473434517198, 10.34069826486997929320441687886, 11.03196722298165007157851563574, 11.66333559236957199128557835438

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