↑

Properties

Label 2-2952-2952.2147-c0-0-1
Degree $2$
Conductor $2952$
Sign $0.880 + 0.474i$
Analytic cond. $1.47323$
Root an. cond. $1.21377$
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  + (0.933 − 0.358i)2-s + (0.913 + 0.406i)3-s + (0.743 − 0.669i)4-s + (0.998 + 0.0523i)6-s + (0.453 − 0.891i)8-s + (0.669 + 0.743i)9-s + (0.287 − 0.811i)11-s + (0.951 − 0.309i)12-s + (0.104 − 0.994i)16-s + (−1.21 − 0.0955i)17-s + (0.891 + 0.453i)18-s + (−0.0273 − 0.0446i)19-s + (−0.0225 − 0.860i)22-s + (0.777 − 0.629i)24-s + (−0.994 − 0.104i)25-s + ⋯
L(s)  = 1  + (0.933 − 0.358i)2-s + (0.913 + 0.406i)3-s + (0.743 − 0.669i)4-s + (0.998 + 0.0523i)6-s + (0.453 − 0.891i)8-s + (0.669 + 0.743i)9-s + (0.287 − 0.811i)11-s + (0.951 − 0.309i)12-s + (0.104 − 0.994i)16-s + (−1.21 − 0.0955i)17-s + (0.891 + 0.453i)18-s + (−0.0273 − 0.0446i)19-s + (−0.0225 − 0.860i)22-s + (0.777 − 0.629i)24-s + (−0.994 − 0.104i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(1.47323\)
Root analytic conductor: \(1.21377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (2147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2952,\ (\ :0),\ 0.880 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.986114499\)
\(L(\frac12)\) \(\approx\) \(2.986114499\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.933 + 0.358i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.838 - 0.544i)T \)
good5 \( 1 + (0.994 + 0.104i)T^{2} \)
7 \( 1 + (0.838 - 0.544i)T^{2} \)
11 \( 1 + (-0.287 + 0.811i)T + (-0.777 - 0.629i)T^{2} \)
13 \( 1 + (0.544 - 0.838i)T^{2} \)
17 \( 1 + (1.21 + 0.0955i)T + (0.987 + 0.156i)T^{2} \)
19 \( 1 + (0.0273 + 0.0446i)T + (-0.453 + 0.891i)T^{2} \)
23 \( 1 + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (0.358 - 0.933i)T^{2} \)
31 \( 1 + (0.913 + 0.406i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.712 - 1.85i)T + (-0.743 + 0.669i)T^{2} \)
47 \( 1 + (-0.0523 + 0.998i)T^{2} \)
53 \( 1 + (0.987 - 0.156i)T^{2} \)
59 \( 1 + (1.92 - 0.201i)T + (0.978 - 0.207i)T^{2} \)
61 \( 1 + (-0.207 + 0.978i)T^{2} \)
67 \( 1 + (-0.656 - 1.85i)T + (-0.777 + 0.629i)T^{2} \)
71 \( 1 + (-0.156 + 0.987i)T^{2} \)
73 \( 1 + (0.147 + 0.147i)T + iT^{2} \)
79 \( 1 + (0.258 + 0.965i)T^{2} \)
83 \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.355 + 1.47i)T + (-0.891 - 0.453i)T^{2} \)
97 \( 1 + (-0.364 + 1.96i)T + (-0.933 - 0.358i)T^{2} \)
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

−9.015746157383436694759605243001, −8.102909383621672649297682003707, −7.37755836632747338609337417472, −6.41473970539453645985167831508, −5.77610278943638041012769083335, −4.60481533429088214537591982331, −4.21339630270190419281174245970, −3.21804089546975813887355590606, −2.57110475446162446467141004655, −1.50690902938799149640202000842, 1.83896618476731090923667401367, 2.41170674623195792307584619483, 3.57480825140419989576421428738, 4.16491054563240736345710336075, 4.99434333549766268383505419487, 6.11785831893479309330247928232, 6.73073440812393253748987807103, 7.43024835257664192192876229029, 7.994030575878359979760465831670, 8.888961074646940320641495164848

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