↑

Properties

Label 2-2925-325.233-c0-0-1
Degree $2$
Conductor $2925$
Sign $0.770 - 0.637i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0712 + 0.139i)2-s + (0.573 + 0.789i)4-s + (−0.972 − 0.233i)5-s + (−0.306 + 0.0484i)8-s + (0.101 − 0.119i)10-s + (0.993 − 0.322i)11-s + (0.891 − 0.453i)13-s + (−0.286 + 0.881i)16-s + (−0.373 − 0.901i)20-s + (−0.0256 + 0.161i)22-s + (0.891 + 0.453i)25-s + 0.156i·26-s + (−0.322 − 0.322i)32-s + (0.309 + 0.0243i)40-s + (1.44 + 0.469i)41-s + ⋯
L(s)  = 1  + (−0.0712 + 0.139i)2-s + (0.573 + 0.789i)4-s + (−0.972 − 0.233i)5-s + (−0.306 + 0.0484i)8-s + (0.101 − 0.119i)10-s + (0.993 − 0.322i)11-s + (0.891 − 0.453i)13-s + (−0.286 + 0.881i)16-s + (−0.373 − 0.901i)20-s + (−0.0256 + 0.161i)22-s + (0.891 + 0.453i)25-s + 0.156i·26-s + (−0.322 − 0.322i)32-s + (0.309 + 0.0243i)40-s + (1.44 + 0.469i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248073556\)
\(L(\frac12)\) \(\approx\) \(1.248073556\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.972 + 0.233i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
good2 \( 1 + (0.0712 - 0.139i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.993 + 0.322i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-1.44 - 0.469i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
47 \( 1 + (1.50 + 0.237i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.144 + 0.444i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (-1.00 - 1.37i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.96 + 0.311i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (-0.570 - 1.75i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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

−8.794803958686770520263066430992, −8.158536610309921983369011314135, −7.70664068754804512571983342319, −6.73038384175583645283299498427, −6.27884910455950588511848673437, −5.13835132872770689583853235803, −3.94130778022021491744938162122, −3.65726783894034355203836011983, −2.62414392359831171879828254290, −1.16777626147925021353299453784, 1.01518617282952363157691675139, 2.09327873108390888069419597577, 3.27533063500999237090205867810, 4.06691076559694813180330791855, 4.89797275471120798940532927052, 6.06613578776081964932640721717, 6.52793943957191206148403919204, 7.28814208723827498219121891722, 8.021795120099369505675987807173, 9.068005980143006175194334897924

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