↑

Properties

Label 2-260-65.4-c1-0-7
Degree $2$
Conductor $260$
Sign $-0.965 - 0.260i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.86 − 1.65i)3-s + (0.877 − 2.05i)5-s + (−0.517 − 0.895i)7-s + (3.96 + 6.86i)9-s + (−2.96 − 1.70i)11-s + (−3.57 + 0.455i)13-s + (−5.91 + 4.43i)15-s + (−2.07 + 1.19i)17-s + (−5.37 + 3.10i)19-s + 3.41i·21-s + (6.28 + 3.62i)23-s + (−3.46 − 3.60i)25-s − 16.3i·27-s + (0.902 − 1.56i)29-s − 5.80i·31-s + ⋯
L(s)  = 1  + (−1.65 − 0.954i)3-s + (0.392 − 0.919i)5-s + (−0.195 − 0.338i)7-s + (1.32 + 2.28i)9-s + (−0.892 − 0.515i)11-s + (−0.992 + 0.126i)13-s + (−1.52 + 1.14i)15-s + (−0.503 + 0.290i)17-s + (−1.23 + 0.711i)19-s + 0.746i·21-s + (1.31 + 0.756i)23-s + (−0.692 − 0.721i)25-s − 3.13i·27-s + (0.167 − 0.290i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.965 - 0.260i$
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: \((2,\ 260,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0422629 + 0.318579i\)
\(L(\frac12)\) \(\approx\) \(0.0422629 + 0.318579i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.877 + 2.05i)T \)
13 \( 1 + (3.57 - 0.455i)T \)
good3 \( 1 + (2.86 + 1.65i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.517 + 0.895i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.96 + 1.70i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.07 - 1.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.37 - 3.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.28 - 3.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.902 + 1.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.80iT - 31T^{2} \)
37 \( 1 + (-0.713 + 1.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.60 + 2.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.86 - 1.07i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.50T + 47T^{2} \)
53 \( 1 + 4.55iT - 53T^{2} \)
59 \( 1 + (5.06 - 2.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.90 + 3.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.80 + 6.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.49 + 5.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.15T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 0.706T + 83T^{2} \)
89 \( 1 + (-5.06 - 2.92i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.99 + 13.8i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56485985055949506071206974960, −10.71994789550245837000059464419, −9.848721658482037881586488540186, −8.325257069396258376543715699713, −7.33257584168053171486773271563, −6.28511735943395282409189902189, −5.43183757781517764060190807795, −4.58830673279857770953107287101, −1.93928100567016866983156915873, −0.29018466030807157641793426373, 2.78261673492602890302361159346, 4.58192183504811074584348513649, 5.27007665067552371159395266131, 6.47206456109769516222321192762, 7.07388070907955516355875027971, 9.053693597758349805491400939141, 10.06950745968034242481675831609, 10.60090875615577188477239826894, 11.27628034027716776452035590017, 12.34483223232978437372253442048

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