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Properties

Label 2-1512-1512.853-c0-0-0
Degree $2$
Conductor $1512$
Sign $0.893 - 0.448i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.866 + 0.5i)3-s + (0.173 + 0.984i)4-s + (0.642 + 0.233i)5-s + (−0.342 − 0.939i)6-s + (−0.173 + 0.984i)7-s + (0.500 − 0.866i)8-s + (0.499 + 0.866i)9-s + (−0.342 − 0.592i)10-s + (−0.342 + 0.939i)12-s + (1.32 − 1.11i)13-s + (0.766 − 0.642i)14-s + (0.439 + 0.524i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)18-s + (−0.984 + 1.70i)19-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.866 + 0.5i)3-s + (0.173 + 0.984i)4-s + (0.642 + 0.233i)5-s + (−0.342 − 0.939i)6-s + (−0.173 + 0.984i)7-s + (0.500 − 0.866i)8-s + (0.499 + 0.866i)9-s + (−0.342 − 0.592i)10-s + (−0.342 + 0.939i)12-s + (1.32 − 1.11i)13-s + (0.766 − 0.642i)14-s + (0.439 + 0.524i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)18-s + (−0.984 + 1.70i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153099500\)
\(L(\frac12)\) \(\approx\) \(1.153099500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
good5 \( 1 + (-0.642 - 0.233i)T + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-1.32 + 1.11i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.223 - 1.26i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-1.32 - 1.11i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−9.782236912148851846402383637855, −8.863602580013889347368695488381, −8.384524459593586565204726890830, −7.88827254356435825516277221546, −6.43577049910179749151960058496, −5.79229740324147060254306533658, −4.33404214275609362332092582009, −3.42652134087575105262950619047, −2.57191074271456775492739513614, −1.77030166451733258046533239232, 1.24093645777323476920213081236, 2.06671495418735154106962551210, 3.59188180894419995581127553469, 4.58615808082370015589672376802, 5.90633339746986738169498162039, 6.63856562771562768456838237350, 7.20129982117782089345442265961, 8.027710948751004668335250477175, 8.934893301833354849922791968616, 9.288522485619064436206004972471

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