Properties

Label 2-1620-5.4-c1-0-16
Degree $2$
Conductor $1620$
Sign $-0.539 + 0.841i$
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  + (−1.20 + 1.88i)5-s + 1.28i·7-s − 4.14·11-s + 6.52i·13-s − 5.98i·17-s − 7.17·19-s − 7.53i·23-s + (−2.08 − 4.54i)25-s + 5.19·29-s + 5.17·31-s + (−2.41 − 1.54i)35-s − 5.24i·37-s − 0.680·41-s − 1.28i·43-s − 5.31i·47-s + ⋯
L(s)  = 1  + (−0.539 + 0.841i)5-s + 0.484i·7-s − 1.24·11-s + 1.80i·13-s − 1.45i·17-s − 1.64·19-s − 1.57i·23-s + (−0.417 − 0.908i)25-s + 0.964·29-s + 0.930·31-s + (−0.407 − 0.261i)35-s − 0.861i·37-s − 0.106·41-s − 0.195i·43-s − 0.774i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.539 + 0.841i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.539 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1735028975\)
\(L(\frac12)\) \(\approx\) \(0.1735028975\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 1.88i)T \)
good7 \( 1 - 1.28iT - 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 - 6.52iT - 13T^{2} \)
17 \( 1 + 5.98iT - 17T^{2} \)
19 \( 1 + 7.17T + 19T^{2} \)
23 \( 1 + 7.53iT - 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 - 5.17T + 31T^{2} \)
37 \( 1 + 5.24iT - 37T^{2} \)
41 \( 1 + 0.680T + 41T^{2} \)
43 \( 1 + 1.28iT - 43T^{2} \)
47 \( 1 + 5.31iT - 47T^{2} \)
53 \( 1 - 2.21iT - 53T^{2} \)
59 \( 1 + 7.60T + 59T^{2} \)
61 \( 1 + 2.17T + 61T^{2} \)
67 \( 1 - 15.6iT - 67T^{2} \)
71 \( 1 + 5.50T + 71T^{2} \)
73 \( 1 + 7.80iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 9.75iT - 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 14.3iT - 97T^{2} \)
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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

−8.920091116850532566557932070839, −8.460799035173657349979676448635, −7.39486148114109378972035465415, −6.76772828578476634070368843942, −6.07040001649963891226836771180, −4.74399401343489623505020707064, −4.23061404336085741059567964309, −2.74231423907335546240935448168, −2.31357399947324902391854163187, −0.06794938029132629590035500001, 1.29008833056833967569872976249, 2.80255466588719061263647788646, 3.80641421395681965773113392101, 4.71038483532351336911112709441, 5.50326168510167161662525843065, 6.32584577155330031416795243644, 7.64640571578020676162580833877, 8.083682044217247318074180447933, 8.544289999485258010686762851813, 9.793751452805722009715750186065

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