↑

Properties

Label 2-460-23.4-c1-0-5
Degree $2$
Conductor $460$
Sign $-0.207 + 0.978i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (−1.08 + 0.319i)3-s + (−0.841 − 0.540i)5-s + (0.0860 + 0.598i)7-s + (−1.44 + 0.927i)9-s + (0.638 − 1.39i)11-s + (0.955 − 6.64i)13-s + (1.08 + 0.319i)15-s + (−1.77 − 2.04i)17-s + (−0.883 + 1.01i)19-s + (−0.284 − 0.623i)21-s + (2.16 − 4.27i)23-s + (0.415 + 0.909i)25-s + (3.50 − 4.03i)27-s + (−4.67 − 5.40i)29-s + (−5.55 − 1.62i)31-s + ⋯
L(s)  = 1  + (−0.627 + 0.184i)3-s + (−0.376 − 0.241i)5-s + (0.0325 + 0.226i)7-s + (−0.480 + 0.309i)9-s + (0.192 − 0.421i)11-s + (0.265 − 1.84i)13-s + (0.280 + 0.0824i)15-s + (−0.430 − 0.496i)17-s + (−0.202 + 0.233i)19-s + (−0.0621 − 0.136i)21-s + (0.452 − 0.891i)23-s + (0.0830 + 0.181i)25-s + (0.673 − 0.777i)27-s + (−0.869 − 1.00i)29-s + (−0.996 − 0.292i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416689 - 0.514107i\)
\(L(\frac12)\) \(\approx\) \(0.416689 - 0.514107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-2.16 + 4.27i)T \)
good3 \( 1 + (1.08 - 0.319i)T + (2.52 - 1.62i)T^{2} \)
7 \( 1 + (-0.0860 - 0.598i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.638 + 1.39i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.955 + 6.64i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.77 + 2.04i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.883 - 1.01i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (4.67 + 5.40i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (5.55 + 1.62i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (6.24 - 4.01i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (6.41 + 4.12i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-12.2 + 3.59i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 1.94T + 47T^{2} \)
53 \( 1 + (1.59 + 11.1i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.88 - 13.0i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-5.45 - 1.60i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (0.362 + 0.794i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-3.21 - 7.03i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (5.25 - 6.06i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (1.07 - 7.48i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (4.24 - 2.72i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (14.1 - 4.14i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (6.87 + 4.41i)T + (40.2 + 88.2i)T^{2} \)
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

−10.87303736705195202358502258732, −10.16865716211173387557584435055, −8.812809410991417934449130964959, −8.226058633259834014131748130367, −7.11679611665467556083480295616, −5.75906111043009121593538619886, −5.33134690351391789727792010072, −3.96560645802709107943490321846, −2.65657531040828441500060452323, −0.44048216078010242182389498330, 1.72004039886468103183849081110, 3.51181887403008649588092454137, 4.53343989713657842210071551061, 5.77313517679348842130949690566, 6.77285594613620970828712757042, 7.33517028346248593815274119002, 8.840741308372315273208717164961, 9.315687516927523463137664030398, 10.84122623288527969639189393080, 11.19526073741554331744523017035

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