↑

Properties

Label 2-1512-24.11-c1-0-18
Degree $2$
Conductor $1512$
Sign $-0.866 - 0.499i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−0.908 + 1.08i)2-s + (−0.347 − 1.96i)4-s + 0.863·5-s + i·7-s + (2.44 + 1.41i)8-s + (−0.784 + 0.935i)10-s + 2.62i·11-s + 1.01i·13-s + (−1.08 − 0.908i)14-s + (−3.75 + 1.36i)16-s + 2.34i·17-s − 4.09·19-s + (−0.300 − 1.70i)20-s + (−2.84 − 2.39i)22-s + 6.23·23-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.173 − 0.984i)4-s + 0.386·5-s + 0.377i·7-s + (0.866 + 0.499i)8-s + (−0.248 + 0.295i)10-s + 0.792i·11-s + 0.282i·13-s + (−0.289 − 0.242i)14-s + (−0.939 + 0.342i)16-s + 0.568i·17-s − 0.939·19-s + (−0.0671 − 0.380i)20-s + (−0.607 − 0.509i)22-s + 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8761871952\)
\(L(\frac12)\) \(\approx\) \(0.8761871952\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.908 - 1.08i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.863T + 5T^{2} \)
11 \( 1 - 2.62iT - 11T^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
19 \( 1 + 4.09T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 + 6.29iT - 31T^{2} \)
37 \( 1 - 5.18iT - 37T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 0.496T + 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 4.40iT - 59T^{2} \)
61 \( 1 - 9.59iT - 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 - 9.44iT - 79T^{2} \)
83 \( 1 - 6.62iT - 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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

−9.688505165008886335868045799451, −8.976271886588918619861582421304, −8.241587496372291061870595058082, −7.42086722889637938794202407306, −6.56627634627830243344746267335, −5.94883801090069220165317860510, −4.99997600739343564880219551072, −4.16051856365152977189514758684, −2.47263752979043014007339716677, −1.47009861454043742926921113971, 0.43462258558654994223710771503, 1.75607297059119590751002838976, 2.89376766993530533529403232369, 3.73893405471473611908945273566, 4.80851474421274156224725684723, 5.86511313586093762261589293181, 6.97543435366151514001460120032, 7.61024582693629142487993533297, 8.738084706647096189464463290290, 8.988685447607747849927857030105

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