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Properties

Label 2-1512-21.20-c1-0-24
Degree $2$
Conductor $1512$
Sign $0.920 + 0.391i$
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  + 2.86·5-s + (2.43 + 1.03i)7-s − 3.32i·11-s − 0.821i·13-s + 3.52·17-s − 5.25i·19-s − 5.12i·23-s + 3.22·25-s + 1.64i·29-s + 2.55i·31-s + (6.98 + 2.96i)35-s − 8.93·37-s + 3.49·41-s − 0.161·43-s + 5.34·47-s + ⋯
L(s)  = 1  + 1.28·5-s + (0.920 + 0.391i)7-s − 1.00i·11-s − 0.227i·13-s + 0.853·17-s − 1.20i·19-s − 1.06i·23-s + 0.645·25-s + 0.305i·29-s + 0.458i·31-s + (1.18 + 0.501i)35-s − 1.46·37-s + 0.546·41-s − 0.0245·43-s + 0.779·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.430702790\)
\(L(\frac12)\) \(\approx\) \(2.430702790\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.43 - 1.03i)T \)
good5 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
13 \( 1 + 0.821iT - 13T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 5.25iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 - 1.64iT - 29T^{2} \)
31 \( 1 - 2.55iT - 31T^{2} \)
37 \( 1 + 8.93T + 37T^{2} \)
41 \( 1 - 3.49T + 41T^{2} \)
43 \( 1 + 0.161T + 43T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 - 3.28iT - 53T^{2} \)
59 \( 1 + 4.08T + 59T^{2} \)
61 \( 1 - 8.43iT - 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 - 7.86iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−9.181462708772673610842000646507, −8.833441562729760988221152700003, −7.936699367452924463398721441084, −6.92144305408143960241507112455, −5.94808058502884280952591409844, −5.43267562871692643264800771050, −4.59659148445740702402656455101, −3.13866641565332157445987505325, −2.24915603918437418344975392197, −1.08795465231508797350559903380, 1.49446583611735210224831392716, 2.04689976877444678177254672938, 3.54279199883309385372885392291, 4.61555824190188058354615851726, 5.46248778347878208677566574365, 6.09146751653275840819234560400, 7.25899467548194270436462650168, 7.78209307404741606545960594870, 8.823954449410544859360060867690, 9.738855401545777927304281124988

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