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Properties

Label 1-260-260.159-r1-0-0
Degree $1$
Conductor $260$
Sign $-0.711 + 0.702i$
Analytic cond. $27.9408$
Root an. cond. $27.9408$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (−0.5 + 0.866i)23-s + 27-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (−0.5 + 0.866i)23-s + 27-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2885759358 + 0.7030505628i\)
\(L(\frac12)\) \(\approx\) \(0.2885759358 + 0.7030505628i\)
\(L(1)\) \(\approx\) \(0.7514693133 + 0.2065287920i\)
\(L(1)\) \(\approx\) \(0.7514693133 + 0.2065287920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.20685841640172445560692351654, −24.49323407029958271606770195032, −23.5062736942998162772401638248, −22.4484849713647457451200210837, −22.136518087079729998872561700602, −20.57557391927709223555199408026, −19.67734447713526802675426032693, −18.668325712753251772266076987569, −18.09675821570934353825854560449, −17.057016991896340208696079720356, −16.11727238304377802194286941454, −15.029439665313085475719028860843, −13.90093586081889649494099060926, −12.85855952549254429633058750783, −12.091374988662927994411119626127, −11.376794591220956877308827698488, −9.92145131698652958594106025386, −8.9529744555456851595967365642, −7.65023351702982654601709012308, −6.74563369633834846275402480002, −5.78739176536956209805376497376, −4.71021982949957055847880718052, −2.92408750837919974313455999420, −1.83803975685569043600676770600, −0.278111294396310026704286469044, 1.20538312935117085435346630322, 3.45153389174854209470425226127, 3.89994548195744032694342757030, 5.44318374104045096751317105357, 6.24171870727706667629892544603, 7.55487821229084681473018456835, 8.90543674038937202679084585922, 9.89612882865615631128414429330, 10.68606755856916890479441581533, 11.604268377501230513311106828265, 12.72757038045922987531295631611, 13.96834955388914816241810959729, 14.77458313331366796222382616899, 16.070821327741838824464125011149, 16.59898712694255980938198671572, 17.37396774671885960045294798791, 18.621054308611217717510200059956, 19.76720807565577522350163900603, 20.50267929661282377156224258902, 21.66089139696240779688799207518, 22.18510526378169346143424979026, 23.31984571978397353618489714361, 23.83897789979334752478204549919, 25.276042629941106826768175522651, 26.19527794669762808745980951008

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