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Properties

Label 2-1512-63.25-c1-0-9
Degree $2$
Conductor $1512$
Sign $0.440 - 0.897i$
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  + (−0.790 + 1.36i)5-s + (2.57 + 0.601i)7-s + (2.58 + 4.47i)11-s + (−0.681 − 1.18i)13-s + (2.30 − 3.99i)17-s + (0.0321 + 0.0557i)19-s + (3.37 − 5.84i)23-s + (1.24 + 2.16i)25-s + (−4.70 + 8.15i)29-s − 2.66·31-s + (−2.86 + 3.05i)35-s + (0.880 + 1.52i)37-s + (0.858 + 1.48i)41-s + (−5.12 + 8.86i)43-s − 5.20·47-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)5-s + (0.973 + 0.227i)7-s + (0.779 + 1.35i)11-s + (−0.189 − 0.327i)13-s + (0.559 − 0.969i)17-s + (0.00738 + 0.0127i)19-s + (0.703 − 1.21i)23-s + (0.249 + 0.432i)25-s + (−0.874 + 1.51i)29-s − 0.478·31-s + (−0.483 + 0.516i)35-s + (0.144 + 0.250i)37-s + (0.134 + 0.232i)41-s + (−0.780 + 1.35i)43-s − 0.759·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.803426941\)
\(L(\frac12)\) \(\approx\) \(1.803426941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 - 0.601i)T \)
good5 \( 1 + (0.790 - 1.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.58 - 4.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.681 + 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.30 + 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0321 - 0.0557i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.37 + 5.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.70 - 8.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.66T + 31T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.858 - 1.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.12 - 8.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.20T + 47T^{2} \)
53 \( 1 + (-0.479 + 0.831i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.33T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + (-5.08 + 8.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.12 - 7.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.26 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.576390644686166850364279384330, −8.865457691388031238566937757170, −7.87902514097231036994613975717, −7.18602916453869732771754518278, −6.64986855003460950522618749933, −5.22688129591597503610393433064, −4.74904617569288313535023323574, −3.59806340780445030770251741292, −2.53822747900402778724843538585, −1.35932665420077311990587576353, 0.812990880631658065892332753687, 1.89845511637555787979052455278, 3.54705956209087372318551549022, 4.12675082510383375332698837320, 5.24514244948395865710436643121, 5.89148261422239342281637055006, 7.00607971960393688509928189441, 7.926496879187966714909723954529, 8.482297623632632681975537732440, 9.138582988526385301226345055015

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