Properties

Label 2-1620-5.3-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.850 + 0.525i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·5-s + (−1 + i)7-s + 11-s + (1 − i)17-s i·19-s + (1 + i)23-s − 25-s i·29-s + 31-s + (1 + i)35-s + 41-s i·49-s + (−1 − i)53-s i·55-s i·59-s + ⋯
L(s)  = 1  i·5-s + (−1 + i)7-s + 11-s + (1 − i)17-s i·19-s + (1 + i)23-s − 25-s i·29-s + 31-s + (1 + i)35-s + 41-s i·49-s + (−1 − i)53-s i·55-s i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083362744\)
\(L(\frac12)\) \(\approx\) \(1.083362744\)
\(L(1)\) 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 \( 1 + iT \)
good7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 - iT^{2} \)
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

−9.429301058329989485656193080175, −8.982538672279099593675159929244, −8.051854261980889296934999516978, −7.06408967374952688612976649064, −6.19403952810484311052916518592, −5.43908222485347726288412255275, −4.63850815168763443828917117720, −3.48768316540448623648721292838, −2.55699935156323982454638098293, −1.03116483750808352984009725253, 1.34082642338976775364110092032, 2.97790725315358273594864002799, 3.59793560300391206916986020241, 4.41326031949971868231633647385, 5.97066965238798052319730917113, 6.42501420608906908551344626600, 7.17467683447287300439154983343, 7.889104379218247201104466059728, 8.983044352910611235262376047170, 9.817285018251890730737094461439

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