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Properties

Label 2-1512-63.4-c1-0-20
Degree $2$
Conductor $1512$
Sign $0.791 + 0.611i$
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  + 1.78·5-s + (1.90 − 1.83i)7-s + 5.61·11-s + (3.14 − 5.43i)13-s + (−0.646 + 1.11i)17-s + (0.559 + 0.968i)19-s − 7.61·23-s − 1.81·25-s + (1.57 + 2.72i)29-s + (−0.501 − 0.868i)31-s + (3.39 − 3.28i)35-s + (−5.96 − 10.3i)37-s + (−4.14 + 7.17i)41-s + (2.34 + 4.06i)43-s + (−0.972 + 1.68i)47-s + ⋯
L(s)  = 1  + 0.797·5-s + (0.718 − 0.695i)7-s + 1.69·11-s + (0.870 − 1.50i)13-s + (−0.156 + 0.271i)17-s + (0.128 + 0.222i)19-s − 1.58·23-s − 0.363·25-s + (0.292 + 0.506i)29-s + (−0.0900 − 0.156i)31-s + (0.573 − 0.554i)35-s + (−0.980 − 1.69i)37-s + (−0.646 + 1.12i)41-s + (0.358 + 0.620i)43-s + (−0.141 + 0.245i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.791 + 0.611i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.360936865\)
\(L(\frac12)\) \(\approx\) \(2.360936865\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.83i)T \)
good5 \( 1 - 1.78T + 5T^{2} \)
11 \( 1 - 5.61T + 11T^{2} \)
13 \( 1 + (-3.14 + 5.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.646 - 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.559 - 0.968i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 + (-1.57 - 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.501 + 0.868i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.96 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.14 - 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.34 - 4.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.972 - 1.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.45 + 7.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.19 - 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.41 - 4.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.27 - 2.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 + 9.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.60 - 2.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.404 - 0.700i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.10 - 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.472748315024764320959266219484, −8.510160165811548158075528235272, −7.931727104836099989796270280961, −6.87904740682891672204694887996, −6.05897109563827152619412333394, −5.43232834020247014550498805715, −4.15082720411367610680564221039, −3.54731322823633359960354959296, −1.96148811533406557735098927552, −1.06989432190566910328151963200, 1.54622067638367013914030553873, 2.08566018548470050834452153874, 3.70591776498940474608107344551, 4.45713810432575429154753694357, 5.56470599603526329764348975074, 6.34873523965007313240851982248, 6.85705724488813974589598522784, 8.200806593310346630349439089159, 8.887002220697947642988788438509, 9.386396811428116157372480045419

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