↑

Properties

Label 2-260-260.127-c1-0-26
Degree $2$
Conductor $260$
Sign $0.990 - 0.141i$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (−0.133 − 2.23i)5-s + (1.99 + 2i)8-s + (2.59 + 1.5i)9-s + (0.633 − 3.09i)10-s + (−3.23 − 1.59i)13-s + (1.99 + 3.46i)16-s + (−0.133 + 0.0358i)17-s + (3 + 3i)18-s + (2 − 3.99i)20-s + (−4.96 + 0.598i)25-s + (−3.83 − 3.36i)26-s + (−9.23 + 5.33i)29-s + (1.46 + 5.46i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (−0.0599 − 0.998i)5-s + (0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (0.200 − 0.979i)10-s + (−0.896 − 0.443i)13-s + (0.499 + 0.866i)16-s + (−0.0324 + 0.00870i)17-s + (0.707 + 0.707i)18-s + (0.447 − 0.894i)20-s + (−0.992 + 0.119i)25-s + (−0.751 − 0.660i)26-s + (−1.71 + 0.989i)29-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.990 - 0.141i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.990 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19611 + 0.155642i\)
\(L(\frac12)\) \(\approx\) \(2.19611 + 0.155642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
13 \( 1 + (3.23 + 1.59i)T \)
good3 \( 1 + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.133 - 0.0358i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (9.23 - 5.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (10.6 + 2.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-10.3 + 5.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-5.29 + 5.29i)T - 53iT^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.16 - 1.16i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.75 + 17.7i)T + (-84.0 + 48.5i)T^{2} \)
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

−12.54160495714783284555854044482, −11.28209905854596109124676965112, −10.25188009445651792846460030184, −9.040700637076031427621710515298, −7.79550374872632366999366220395, −7.11521259470731472799974629183, −5.56958272668486288646026675146, −4.86561018704451972422094923930, −3.76036702768594012231891319056, −1.95942638812661455191096502919, 2.08898932961594158668462306208, 3.46000644685835524707585614927, 4.49210578042835974567409388455, 5.89933104727720344581735926307, 6.91142190369548993287531999010, 7.55302108494919958138867422268, 9.509050851451376614445355339151, 10.21471240296305693964805836261, 11.22008427892027977458233359061, 11.98021849873678474732146875670

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