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Properties

Label 2-260-65.4-c1-0-1
Degree $2$
Conductor $260$
Sign $0.487 - 0.872i$
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  + (1.28 + 0.739i)3-s + (0.494 + 2.18i)5-s + (1.56 + 2.71i)7-s + (−0.406 − 0.704i)9-s + (−4.01 − 2.31i)11-s + (−2.44 + 2.64i)13-s + (−0.979 + 3.15i)15-s + (5.87 − 3.38i)17-s + (1.45 − 0.839i)19-s + 4.63i·21-s + (4.79 + 2.76i)23-s + (−4.51 + 2.15i)25-s − 5.63i·27-s + (3.87 − 6.70i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (0.739 + 0.426i)3-s + (0.220 + 0.975i)5-s + (0.592 + 1.02i)7-s + (−0.135 − 0.234i)9-s + (−1.20 − 0.698i)11-s + (−0.678 + 0.734i)13-s + (−0.252 + 0.815i)15-s + (1.42 − 0.821i)17-s + (0.333 − 0.192i)19-s + 1.01i·21-s + (0.999 + 0.576i)23-s + (−0.902 + 0.431i)25-s − 1.08i·27-s + (0.718 − 1.24i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.487 - 0.872i$
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: \((2,\ 260,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37708 + 0.807941i\)
\(L(\frac12)\) \(\approx\) \(1.37708 + 0.807941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.494 - 2.18i)T \)
13 \( 1 + (2.44 - 2.64i)T \)
good3 \( 1 + (-1.28 - 0.739i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.56 - 2.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.01 + 2.31i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.87 + 3.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 0.839i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.79 - 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.87 + 6.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-3.72 + 6.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.78 - 2.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.7 - 6.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.97T + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (-2.27 + 1.31i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.87 + 8.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.317 - 0.550i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.0 - 6.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 - 3.33T + 83T^{2} \)
89 \( 1 + (2.27 + 1.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.13 - 5.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.85423363661143291881108983368, −11.33285005438595275246725133523, −10.01376977077373152380561606512, −9.402596539298581102258109828644, −8.283072318639036894848256295700, −7.43759888736132882097175982164, −5.99755752478803032633209359646, −5.00723978744851734371439121063, −3.19616979682000423996245737328, −2.53466254367154156400357350346, 1.39072247708684341702378206846, 2.95750305763899889422686126512, 4.67637767210095674464865110032, 5.43018257823803435146974441768, 7.36992158883999339249092018788, 7.86955921468376622349740892216, 8.661974364525970554677716200472, 10.08056145587231009664075544202, 10.57112297919373760889933872243, 12.15168275939722868203589888256

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