Properties

Label 2-1512-24.11-c1-0-88
Degree $2$
Conductor $1512$
Sign $-0.643 + 0.765i$
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  + (0.971 − 1.02i)2-s + (−0.114 − 1.99i)4-s + 2.21·5-s i·7-s + (−2.16 − 1.82i)8-s + (2.14 − 2.27i)10-s − 2.42i·11-s − 1.11i·13-s + (−1.02 − 0.971i)14-s + (−3.97 + 0.456i)16-s − 0.701i·17-s + 0.938·19-s + (−0.252 − 4.41i)20-s + (−2.49 − 2.35i)22-s + 4.30·23-s + ⋯
L(s)  = 1  + (0.686 − 0.727i)2-s + (−0.0571 − 0.998i)4-s + 0.989·5-s − 0.377i·7-s + (−0.765 − 0.643i)8-s + (0.679 − 0.719i)10-s − 0.730i·11-s − 0.309i·13-s + (−0.274 − 0.259i)14-s + (−0.993 + 0.114i)16-s − 0.170i·17-s + 0.215·19-s + (−0.0565 − 0.988i)20-s + (−0.531 − 0.501i)22-s + 0.897·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.643 + 0.765i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.671954039\)
\(L(\frac12)\) \(\approx\) \(2.671954039\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.971 + 1.02i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 2.21T + 5T^{2} \)
11 \( 1 + 2.42iT - 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 0.701iT - 17T^{2} \)
19 \( 1 - 0.938T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 + 4.26T + 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 + 0.157iT - 41T^{2} \)
43 \( 1 - 7.08T + 43T^{2} \)
47 \( 1 + 0.867T + 47T^{2} \)
53 \( 1 + 3.91T + 53T^{2} \)
59 \( 1 + 7.28iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 - 6.65iT - 79T^{2} \)
83 \( 1 + 8.41iT - 83T^{2} \)
89 \( 1 - 7.34iT - 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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

−9.579979246182600311879807282879, −8.620537726454117796873368307066, −7.45915189501813518969158245927, −6.41568699526754509939366878124, −5.75270276846335844726199080424, −5.06167656941737574841989359918, −3.96641637233095976302101685827, −3.03098474525759442385435000929, −2.05420690578821436280402188741, −0.836216005303567950198516573384, 1.83758478918113025153664333576, 2.83074583128789796761389420484, 3.98768052172124422233184832529, 5.02576699701692671864436967753, 5.59039725257036146454707592900, 6.47137436124757562469151878396, 7.13338734364673507796244633226, 8.012324435267751900681790479619, 9.070490306664945429728536688457, 9.426408480638858960385438204021

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