Properties

Label 2-260-65.63-c1-0-1
Degree $2$
Conductor $260$
Sign $0.863 - 0.503i$
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  + (0.133 − 0.5i)3-s + (−1 + 2i)5-s + (−0.232 − 0.133i)7-s + (2.36 + 1.36i)9-s + (4.59 + 1.23i)11-s + (3 + 2i)13-s + (0.866 + 0.767i)15-s + (−2.86 + 0.767i)17-s + (−0.866 − 3.23i)19-s + (−0.0980 + 0.0980i)21-s + (−0.133 − 0.0358i)23-s + (−3 − 4i)25-s + (2.09 − 2.09i)27-s + (−1.03 + 0.598i)29-s + (2.26 + 2.26i)31-s + ⋯
L(s)  = 1  + (0.0773 − 0.288i)3-s + (−0.447 + 0.894i)5-s + (−0.0877 − 0.0506i)7-s + (0.788 + 0.455i)9-s + (1.38 + 0.371i)11-s + (0.832 + 0.554i)13-s + (0.223 + 0.198i)15-s + (−0.695 + 0.186i)17-s + (−0.198 − 0.741i)19-s + (−0.0214 + 0.0214i)21-s + (−0.0279 − 0.00748i)23-s + (−0.600 − 0.800i)25-s + (0.403 − 0.403i)27-s + (−0.192 + 0.111i)29-s + (0.407 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.863 - 0.503i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26465 + 0.341702i\)
\(L(\frac12)\) \(\approx\) \(1.26465 + 0.341702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
13 \( 1 + (-3 - 2i)T \)
good3 \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.232 + 0.133i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.59 - 1.23i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.86 - 0.767i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.866 + 3.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.133 + 0.0358i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.03 - 0.598i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.26 - 2.26i)T + 31iT^{2} \)
37 \( 1 + (3.23 - 1.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 6.96i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.66 + 9.96i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.46iT - 47T^{2} \)
53 \( 1 + (8.46 + 8.46i)T + 53iT^{2} \)
59 \( 1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.86 + 0.767i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (0.794 - 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.23 + 2.13i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.90341736566227359578056214058, −11.21056646571931701769983121372, −10.27429989412980363624660931575, −9.182068357785622667546952044579, −8.107498813846741626312997285450, −6.74971256443723792136709754852, −6.64788167196521563928874840225, −4.58193133376093330025124441029, −3.57335143420865868677238473167, −1.83917472684112210544127837166, 1.25771307979135068966093774660, 3.64456328275796076045641318957, 4.35584187622343341792790672418, 5.81953960837875859378702042148, 6.90604809138374494331939667463, 8.252047313603479156648744273117, 9.000117031525289714409109734392, 9.835633949759491437321759689945, 11.06946078957426409823089273629, 11.96147327865989119037248429661

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