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Properties

Label 2-1512-24.11-c1-0-74
Degree $2$
Conductor $1512$
Sign $-0.603 + 0.797i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 − 0.952i)2-s + (0.183 + 1.99i)4-s + 2.35·5-s + i·7-s + (1.70 − 2.25i)8-s + (−2.45 − 2.23i)10-s − 1.09i·11-s − 4.10i·13-s + (0.952 − 1.04i)14-s + (−3.93 + 0.731i)16-s − 4.74i·17-s − 7.10·19-s + (0.431 + 4.68i)20-s + (−1.03 + 1.13i)22-s − 3.99·23-s + ⋯
L(s)  = 1  + (−0.738 − 0.673i)2-s + (0.0918 + 0.995i)4-s + 1.05·5-s + 0.377i·7-s + (0.603 − 0.797i)8-s + (−0.776 − 0.708i)10-s − 0.328i·11-s − 1.13i·13-s + (0.254 − 0.279i)14-s + (−0.983 + 0.182i)16-s − 1.15i·17-s − 1.62·19-s + (0.0965 + 1.04i)20-s + (−0.221 + 0.243i)22-s − 0.832·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.603 + 0.797i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.010370643\)
\(L(\frac12)\) \(\approx\) \(1.010370643\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 + 0.952i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 2.35T + 5T^{2} \)
11 \( 1 + 1.09iT - 11T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
17 \( 1 + 4.74iT - 17T^{2} \)
19 \( 1 + 7.10T + 19T^{2} \)
23 \( 1 + 3.99T + 23T^{2} \)
29 \( 1 - 4.09T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 + 1.29iT - 41T^{2} \)
43 \( 1 + 8.59T + 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 + 0.688T + 53T^{2} \)
59 \( 1 + 7.72iT - 59T^{2} \)
61 \( 1 + 9.70iT - 61T^{2} \)
67 \( 1 - 8.23T + 67T^{2} \)
71 \( 1 - 6.46T + 71T^{2} \)
73 \( 1 - 8.44T + 73T^{2} \)
79 \( 1 - 5.70iT - 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 - 2.54iT - 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.520842473577280877510412549143, −8.374422131555300457776385509520, −8.016156231971770033752733660743, −6.75120346968126619956937294800, −6.01667704771073039656601188411, −5.03787769499861444346655325962, −3.83195688916912424751535210657, −2.64451356544828827782614288216, −2.04952590535250770747016655143, −0.48882438639891028416756056985, 1.53118519891884346063861401019, 2.20200021680646054174773068338, 4.04889721976497982803142263591, 4.92349057364708316975425561000, 6.04622625330929937369521507994, 6.47985863504716158085081717511, 7.22242772341180733302274411724, 8.416592005054865653537664953839, 8.784774547778639431205076335474, 9.768896643990106131661305415843

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