Properties

Label 2-1620-45.14-c2-0-30
Degree $2$
Conductor $1620$
Sign $0.787 + 0.615i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.45 − 2.26i)5-s + (−2.97 + 1.71i)7-s + (11.1 − 6.41i)11-s + (1.55 + 0.896i)13-s + 30.9·17-s − 19.2·19-s + (1.19 − 2.06i)23-s + (14.6 − 20.2i)25-s + (−31.0 + 17.9i)29-s + (20.4 − 35.4i)31-s + (−9.36 + 14.4i)35-s + 53.6i·37-s + (2.14 + 1.24i)41-s + (47.7 − 27.5i)43-s + (−28.5 − 49.4i)47-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)5-s + (−0.425 + 0.245i)7-s + (1.00 − 0.582i)11-s + (0.119 + 0.0689i)13-s + 1.82·17-s − 1.01·19-s + (0.0518 − 0.0897i)23-s + (0.587 − 0.808i)25-s + (−1.07 + 0.618i)29-s + (0.659 − 1.14i)31-s + (−0.267 + 0.411i)35-s + 1.44i·37-s + (0.0524 + 0.0302i)41-s + (1.11 − 0.641i)43-s + (−0.606 − 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.787 + 0.615i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.787 + 0.615i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.535277167\)
\(L(\frac12)\) \(\approx\) \(2.535277167\)
\(L(2)\) 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 \( 1 + (-4.45 + 2.26i)T \)
good7 \( 1 + (2.97 - 1.71i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-11.1 + 6.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.55 - 0.896i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 30.9T + 289T^{2} \)
19 \( 1 + 19.2T + 361T^{2} \)
23 \( 1 + (-1.19 + 2.06i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (31.0 - 17.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-20.4 + 35.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 53.6iT - 1.36e3T^{2} \)
41 \( 1 + (-2.14 - 1.24i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-47.7 + 27.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (28.5 + 49.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.4T + 2.80e3T^{2} \)
59 \( 1 + (-59.8 - 34.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.80 + 15.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-44.3 - 25.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.9iT - 5.04e3T^{2} \)
73 \( 1 + 42.7iT - 5.32e3T^{2} \)
79 \( 1 + (44.4 + 76.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (14.0 + 24.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 68.3iT - 7.92e3T^{2} \)
97 \( 1 + (-135. + 78.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.107483242427058461116895659066, −8.549866471625708371576407414198, −7.56962551368056212900843104140, −6.41277039708573721402516619626, −5.99016168435774398637576791872, −5.15134034914151177828722911990, −4.01146707185136500546611423358, −3.08343433886416834897475899308, −1.85630565807924285015493122726, −0.810837174099772088918306106262, 1.09599882342801015517260907025, 2.16381134749386164708930701013, 3.30552511768430263205332162707, 4.13597424696388616638616679023, 5.37272549612005156032518426397, 6.09417351026819000389182032001, 6.82219627425425893625282880246, 7.55076919160221613740089459189, 8.600746098854772530484714948413, 9.592698876336951026184842168970

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