Properties

Label 2-1470-7.6-c2-0-27
Degree $2$
Conductor $1470$
Sign $0.755 - 0.654i$
Analytic cond. $40.0545$
Root an. cond. $6.32887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.23i·5-s − 2.44i·6-s − 2.82·8-s − 2.99·9-s − 3.16i·10-s + 12.3·11-s + 3.46i·12-s − 7.26i·13-s − 3.87·15-s + 4.00·16-s + 9.29i·17-s + 4.24·18-s − 6.07i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.447i·5-s − 0.408i·6-s − 0.353·8-s − 0.333·9-s − 0.316i·10-s + 1.12·11-s + 0.288i·12-s − 0.558i·13-s − 0.258·15-s + 0.250·16-s + 0.546i·17-s + 0.235·18-s − 0.319i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(40.0545\)
Root analytic conductor: \(6.32887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.489277888\)
\(L(\frac12)\) \(\approx\) \(1.489277888\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 \)
good11 \( 1 - 12.3T + 121T^{2} \)
13 \( 1 + 7.26iT - 169T^{2} \)
17 \( 1 - 9.29iT - 289T^{2} \)
19 \( 1 + 6.07iT - 361T^{2} \)
23 \( 1 + 2.25T + 529T^{2} \)
29 \( 1 - 42.2T + 841T^{2} \)
31 \( 1 + 1.21iT - 961T^{2} \)
37 \( 1 - 35.0T + 1.36e3T^{2} \)
41 \( 1 + 57.8iT - 1.68e3T^{2} \)
43 \( 1 + 34.0T + 1.84e3T^{2} \)
47 \( 1 + 57.0iT - 2.20e3T^{2} \)
53 \( 1 - 14.5T + 2.80e3T^{2} \)
59 \( 1 - 57.8iT - 3.48e3T^{2} \)
61 \( 1 - 5.86iT - 3.72e3T^{2} \)
67 \( 1 - 49.4T + 4.48e3T^{2} \)
71 \( 1 - 101.T + 5.04e3T^{2} \)
73 \( 1 - 82.3iT - 5.32e3T^{2} \)
79 \( 1 - 111.T + 6.24e3T^{2} \)
83 \( 1 + 91.6iT - 6.88e3T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 + 61.4iT - 9.40e3T^{2} \)
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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

−9.414857968197399374243755693859, −8.697479318150306049911625912183, −8.006352761191812542143715185091, −6.94575446101481859321590465793, −6.33239973002316806764981236491, −5.36147828133308165403181417782, −4.17021208810997086021297683950, −3.32573742960733937159223126023, −2.22037912606051690024197590401, −0.806394637038519520610968556288, 0.797116461268831701971172633050, 1.66071112405278113036678299744, 2.84957434402701604861660661179, 4.09480514285322376402732201484, 5.12637105516614207321025601900, 6.43839103735648340057758371842, 6.63125827504329959250157651281, 7.85114963326218129577020033287, 8.329400255707159438146288793071, 9.349519049795569812115097984527

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