Properties

Label 2-1620-15.14-c2-0-27
Degree $2$
Conductor $1620$
Sign $0.556 + 0.830i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (−2.78 − 4.15i)5-s − 0.701i·7-s − 0.462i·11-s + 14.1i·13-s − 14.3·17-s + 5.37·19-s + 41.6·23-s + (−9.51 + 23.1i)25-s − 9.34i·29-s + 21.8·31-s + (−2.91 + 1.95i)35-s + 34.0i·37-s − 22.3i·41-s − 55.9i·43-s + 7.64·47-s + ⋯
L(s)  = 1  + (−0.556 − 0.830i)5-s − 0.100i·7-s − 0.0420i·11-s + 1.08i·13-s − 0.846·17-s + 0.283·19-s + 1.81·23-s + (−0.380 + 0.924i)25-s − 0.322i·29-s + 0.705·31-s + (−0.0833 + 0.0558i)35-s + 0.921i·37-s − 0.545i·41-s − 1.30i·43-s + 0.162·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.556 + 0.830i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.574639256\)
\(L(\frac12)\) \(\approx\) \(1.574639256\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.78 + 4.15i)T \)
good7 \( 1 + 0.701iT - 49T^{2} \)
11 \( 1 + 0.462iT - 121T^{2} \)
13 \( 1 - 14.1iT - 169T^{2} \)
17 \( 1 + 14.3T + 289T^{2} \)
19 \( 1 - 5.37T + 361T^{2} \)
23 \( 1 - 41.6T + 529T^{2} \)
29 \( 1 + 9.34iT - 841T^{2} \)
31 \( 1 - 21.8T + 961T^{2} \)
37 \( 1 - 34.0iT - 1.36e3T^{2} \)
41 \( 1 + 22.3iT - 1.68e3T^{2} \)
43 \( 1 + 55.9iT - 1.84e3T^{2} \)
47 \( 1 - 7.64T + 2.20e3T^{2} \)
53 \( 1 - 36.2T + 2.80e3T^{2} \)
59 \( 1 + 35.0iT - 3.48e3T^{2} \)
61 \( 1 + 58.8T + 3.72e3T^{2} \)
67 \( 1 - 61.8iT - 4.48e3T^{2} \)
71 \( 1 + 43.6iT - 5.04e3T^{2} \)
73 \( 1 + 50.7iT - 5.32e3T^{2} \)
79 \( 1 + 96.1T + 6.24e3T^{2} \)
83 \( 1 + 26.0T + 6.88e3T^{2} \)
89 \( 1 + 125. iT - 7.92e3T^{2} \)
97 \( 1 + 117. iT - 9.40e3T^{2} \)
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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

−8.842603713035218766022530648174, −8.604738917665492329492466162805, −7.36692428191597483700356475202, −6.87907369786959283392536130194, −5.74226417202208332339433651144, −4.74674682878995638327710558012, −4.23475498379602944697118359952, −3.09156007584488427452202100240, −1.76703788350375609238217281573, −0.57564484538035185790535747838, 0.875253691115500893813222898812, 2.55520901391792703999781709905, 3.18798985355806012323463135730, 4.26708407797805704491817302333, 5.21272192466764670694753226779, 6.19809357180323068479368311017, 7.02172712580857441115836084026, 7.64388161849244642374893699642, 8.496755826166832043412463374033, 9.287875662559289864715544640810

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