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Properties

Label 2-1620-9.4-c1-0-3
Degree $2$
Conductor $1620$
Sign $-0.173 - 0.984i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
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  + (−0.5 + 0.866i)5-s + (2 + 3.46i)7-s + (3 + 5.19i)11-s + (2 − 3.46i)13-s + 3·17-s − 7·19-s + (−4.5 + 7.79i)23-s + (−0.499 − 0.866i)25-s + (3.5 − 6.06i)31-s − 3.99·35-s + 2·37-s + (3 − 5.19i)41-s + (−1 − 1.73i)43-s + (−4.49 + 7.79i)49-s − 9·53-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.755 + 1.30i)7-s + (0.904 + 1.56i)11-s + (0.554 − 0.960i)13-s + 0.727·17-s − 1.60·19-s + (−0.938 + 1.62i)23-s + (−0.0999 − 0.173i)25-s + (0.628 − 1.08i)31-s − 0.676·35-s + 0.328·37-s + (0.468 − 0.811i)41-s + (−0.152 − 0.264i)43-s + (−0.642 + 1.11i)49-s − 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.700553425\)
\(L(\frac12)\) \(\approx\) \(1.700553425\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.599073670647875580890225106251, −8.797360495766129746582394319092, −7.986123386515366497756441329888, −7.39802598763511752249860552070, −6.20998983155511659215035609453, −5.67840991366308643755309491143, −4.58863493562155675783775664763, −3.75920651264506611501649039411, −2.45650225168527685564653888210, −1.63395255269694657377759452445, 0.68854983713981989895663223638, 1.69144646155458567491023220668, 3.37562301298493495889530424173, 4.20470730540135368422914554810, 4.71475621974649754560240220270, 6.30165999614062061506762570897, 6.45904092928053239715152338579, 7.80960608491465966794122101785, 8.382808113225048940027046662983, 8.925267328207665251863436388048

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