Properties

Label 2-1620-15.14-c2-0-38
Degree $2$
Conductor $1620$
Sign $-0.838 + 0.544i$
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  + (4.19 − 2.72i)5-s + 3.43i·7-s + 12.8i·11-s + 1.79i·13-s − 30.9·17-s − 19.2·19-s + 2.38·23-s + (10.1 − 22.8i)25-s − 35.8i·29-s − 40.9·31-s + (9.36 + 14.4i)35-s − 53.6i·37-s − 2.48i·41-s − 55.1i·43-s − 57.0·47-s + ⋯
L(s)  = 1  + (0.838 − 0.544i)5-s + 0.491i·7-s + 1.16i·11-s + 0.137i·13-s − 1.82·17-s − 1.01·19-s + 0.103·23-s + (0.406 − 0.913i)25-s − 1.23i·29-s − 1.31·31-s + (0.267 + 0.411i)35-s − 1.44i·37-s − 0.0605i·41-s − 1.28i·43-s − 1.21·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.838 + 0.544i$
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: \((2,\ 1620,\ (\ :1),\ -0.838 + 0.544i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5491275965\)
\(L(\frac12)\) \(\approx\) \(0.5491275965\)
\(L(2)\) 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 \( 1 + (-4.19 + 2.72i)T \)
good7 \( 1 - 3.43iT - 49T^{2} \)
11 \( 1 - 12.8iT - 121T^{2} \)
13 \( 1 - 1.79iT - 169T^{2} \)
17 \( 1 + 30.9T + 289T^{2} \)
19 \( 1 + 19.2T + 361T^{2} \)
23 \( 1 - 2.38T + 529T^{2} \)
29 \( 1 + 35.8iT - 841T^{2} \)
31 \( 1 + 40.9T + 961T^{2} \)
37 \( 1 + 53.6iT - 1.36e3T^{2} \)
41 \( 1 + 2.48iT - 1.68e3T^{2} \)
43 \( 1 + 55.1iT - 1.84e3T^{2} \)
47 \( 1 + 57.0T + 2.20e3T^{2} \)
53 \( 1 + 19.4T + 2.80e3T^{2} \)
59 \( 1 + 69.1iT - 3.48e3T^{2} \)
61 \( 1 - 17.6T + 3.72e3T^{2} \)
67 \( 1 - 51.1iT - 4.48e3T^{2} \)
71 \( 1 + 53.9iT - 5.04e3T^{2} \)
73 \( 1 - 42.7iT - 5.32e3T^{2} \)
79 \( 1 - 88.8T + 6.24e3T^{2} \)
83 \( 1 + 28.0T + 6.88e3T^{2} \)
89 \( 1 + 68.3iT - 7.92e3T^{2} \)
97 \( 1 + 156. iT - 9.40e3T^{2} \)
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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

−9.047680296684922697508840568575, −8.288374776289678286648376935928, −7.13935296035195453275321395309, −6.45197438253236502851142082061, −5.59599839773856257602934502489, −4.72261186903755967095270622828, −4.01355328096929777095269066581, −2.20464028184857417649319981506, −2.02957862853434421444033065412, −0.13277000108998630384213170689, 1.44238136527174968068956407975, 2.56563844252817976975808875438, 3.48422281201705242314071576414, 4.57231202965010094235190786801, 5.54261318904817937335503700393, 6.51015934811610196972343081263, 6.80350452761544077559162650002, 8.031081733503581052510702110271, 8.836186807506411163381612556518, 9.429921347291733146228052406797

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