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Properties

Label 2-1512-1512.1357-c0-0-2
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  + (0.939 − 0.342i)2-s + (0.866 + 0.5i)3-s + (0.766 − 0.642i)4-s + (0.342 + 1.93i)5-s + (0.984 + 0.173i)6-s + (−0.766 − 0.642i)7-s + (0.500 − 0.866i)8-s + (0.499 + 0.866i)9-s + (0.984 + 1.70i)10-s + (0.984 − 0.173i)12-s + (−1.62 − 0.592i)13-s + (−0.939 − 0.342i)14-s + (−0.673 + 1.85i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)18-s + (0.642 − 1.11i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.866 + 0.5i)3-s + (0.766 − 0.642i)4-s + (0.342 + 1.93i)5-s + (0.984 + 0.173i)6-s + (−0.766 − 0.642i)7-s + (0.500 − 0.866i)8-s + (0.499 + 0.866i)9-s + (0.984 + 1.70i)10-s + (0.984 − 0.173i)12-s + (−1.62 − 0.592i)13-s + (−0.939 − 0.342i)14-s + (−0.673 + 1.85i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)18-s + (0.642 − 1.11i)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: $\chi_{1512} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.404662956\)
\(L(\frac12)\) \(\approx\) \(2.404662956\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (-0.342 - 1.93i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.939 + 0.342i)T^{2} \)
13 \( 1 + (1.62 + 0.592i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.524 + 0.439i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (1.62 - 0.592i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−9.788177874426393533096756586725, −9.539117986252588293652274340210, −7.69577754686174650105969468326, −7.15465844218236917014670059842, −6.65635238686115830479551499206, −5.50166906589584644568729272199, −4.48650949402302713580477184581, −3.40547587770143503639932673827, −2.91378721451946261891682757203, −2.29407987741133748094483266886, 1.64138769600830376752152863568, 2.54295875971935912507878528737, 3.72148626139963056616502696368, 4.66032294720260173370064505812, 5.41356120076088139181917356027, 6.19648011039595040952791595061, 7.24893500274497615158763987629, 7.966167295055421539863418681705, 8.743449370710842083558591295522, 9.423894561128427828337162757425

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