Properties

Label 2-460-115.104-c1-0-2
Degree $2$
Conductor $460$
Sign $0.160 - 0.987i$
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  + (−2.38 − 0.343i)3-s + (2.04 + 0.893i)5-s + (−0.826 + 0.716i)7-s + (2.69 + 0.791i)9-s + (−1.07 − 0.688i)11-s + (0.352 + 0.305i)13-s + (−4.58 − 2.83i)15-s + (−0.504 + 0.230i)17-s + (−2.60 + 5.69i)19-s + (2.21 − 1.42i)21-s + (1.37 + 4.59i)23-s + (3.40 + 3.66i)25-s + (0.417 + 0.190i)27-s + (3.77 + 8.27i)29-s + (0.476 + 3.31i)31-s + ⋯
L(s)  = 1  + (−1.37 − 0.198i)3-s + (0.916 + 0.399i)5-s + (−0.312 + 0.270i)7-s + (0.898 + 0.263i)9-s + (−0.322 − 0.207i)11-s + (0.0978 + 0.0847i)13-s + (−1.18 − 0.731i)15-s + (−0.122 + 0.0558i)17-s + (−0.596 + 1.30i)19-s + (0.484 − 0.311i)21-s + (0.285 + 0.958i)23-s + (0.680 + 0.732i)25-s + (0.0803 + 0.0367i)27-s + (0.701 + 1.53i)29-s + (0.0856 + 0.595i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601662 + 0.511649i\)
\(L(\frac12)\) \(\approx\) \(0.601662 + 0.511649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 0.893i)T \)
23 \( 1 + (-1.37 - 4.59i)T \)
good3 \( 1 + (2.38 + 0.343i)T + (2.87 + 0.845i)T^{2} \)
7 \( 1 + (0.826 - 0.716i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.07 + 0.688i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-0.352 - 0.305i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.504 - 0.230i)T + (11.1 - 12.8i)T^{2} \)
19 \( 1 + (2.60 - 5.69i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-3.77 - 8.27i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.476 - 3.31i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-0.601 + 2.04i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.622 - 0.182i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.451 - 0.0649i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 7.73iT - 47T^{2} \)
53 \( 1 + (8.34 - 7.23i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (5.65 - 6.53i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.781 + 5.43i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (0.346 + 0.539i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-6.97 + 4.47i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-3.75 - 1.71i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-2.51 + 2.89i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (0.460 - 1.56i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.33 + 9.28i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (0.295 + 1.00i)T + (-81.6 + 52.4i)T^{2} \)
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

−11.03666836097774598714649580957, −10.60938604931552446643279600607, −9.713446754435994448090403938198, −8.653601795339726050226714024803, −7.26015231738801389357672136889, −6.33279358170460498578370156194, −5.78347395447759378428251463679, −4.91530171481478728459489349945, −3.20343866272950216164563479270, −1.56138923104757562033973826654, 0.59537575271773031508199119925, 2.48883155112028960898098970634, 4.46271422173558794150778709796, 5.10836422051529880861486227631, 6.23351032807787454020929917027, 6.65656885208526383660417655729, 8.172311441672810644283721959512, 9.332572281052281791102669243986, 10.12227041600321976740781833037, 10.82582221594825323066899928550

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