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Properties

Label 2-1512-8.5-c1-0-49
Degree $2$
Conductor $1512$
Sign $0.890 - 0.455i$
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  + (−1.09 + 0.890i)2-s + (0.414 − 1.95i)4-s + 1.58i·5-s + 7-s + (1.28 + 2.51i)8-s + (−1.41 − 1.74i)10-s − 0.790i·11-s − 0.494i·13-s + (−1.09 + 0.890i)14-s + (−3.65 − 1.62i)16-s + 4.46·17-s − 7.55i·19-s + (3.10 + 0.656i)20-s + (0.703 + 0.868i)22-s + 0.839·23-s + ⋯
L(s)  = 1  + (−0.776 + 0.629i)2-s + (0.207 − 0.978i)4-s + 0.708i·5-s + 0.377·7-s + (0.455 + 0.890i)8-s + (−0.446 − 0.550i)10-s − 0.238i·11-s − 0.137i·13-s + (−0.293 + 0.237i)14-s + (−0.914 − 0.405i)16-s + 1.08·17-s − 1.73i·19-s + (0.693 + 0.146i)20-s + (0.150 + 0.185i)22-s + 0.175·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.217254371\)
\(L(\frac12)\) \(\approx\) \(1.217254371\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.890i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.58iT - 5T^{2} \)
11 \( 1 + 0.790iT - 11T^{2} \)
13 \( 1 + 0.494iT - 13T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
19 \( 1 + 7.55iT - 19T^{2} \)
23 \( 1 - 0.839T + 23T^{2} \)
29 \( 1 - 2.76iT - 29T^{2} \)
31 \( 1 + 0.568T + 31T^{2} \)
37 \( 1 + 0.343iT - 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 + 4.36iT - 53T^{2} \)
59 \( 1 + 4.50iT - 59T^{2} \)
61 \( 1 + 5.40iT - 61T^{2} \)
67 \( 1 - 7.57iT - 67T^{2} \)
71 \( 1 - 9.52T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 3.71T + 79T^{2} \)
83 \( 1 - 7.30iT - 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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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

−9.432148077096854899040885520812, −8.683578247460112011300140558253, −7.927153126570196856697119897697, −7.07224952284463758324470127472, −6.61627761690935475686303122261, −5.48836548957009352703018169845, −4.85800837491507989878725773355, −3.35324960196082000200118854749, −2.27144065221365105682653189061, −0.826206709447084065390081932356, 1.01531600295828162404527700018, 1.93735582143939808864064975445, 3.26715410140946721044954732705, 4.16692995098946129262782103102, 5.16032890998597531019892717067, 6.20341119556715247536092673611, 7.38281921232779072952887866629, 8.001083132952060958851067725868, 8.635652700889697464198422948944, 9.474134904924729921818373853791

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