Properties

Label 2-1620-5.4-c1-0-17
Degree $2$
Conductor $1620$
Sign $0.0552 + 0.998i$
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  + (0.123 + 2.23i)5-s − 1.28i·7-s − 5.09·11-s − 3.56i·13-s − 0.895i·17-s + 5.34·19-s − 5.06i·23-s + (−4.96 + 0.551i)25-s − 3·29-s − 6.59·31-s + (2.87 − 0.159i)35-s − 7.24i·37-s + 7.84·41-s − 10.9i·43-s − 2.96i·47-s + ⋯
L(s)  = 1  + (0.0552 + 0.998i)5-s − 0.486i·7-s − 1.53·11-s − 0.990i·13-s − 0.217i·17-s + 1.22·19-s − 1.05i·23-s + (−0.993 + 0.110i)25-s − 0.557·29-s − 1.18·31-s + (0.486 − 0.0269i)35-s − 1.19i·37-s + 1.22·41-s − 1.66i·43-s − 0.433i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0552 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0552 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.0552 + 0.998i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.0552 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.018039421\)
\(L(\frac12)\) \(\approx\) \(1.018039421\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.123 - 2.23i)T \)
good7 \( 1 + 1.28iT - 7T^{2} \)
11 \( 1 + 5.09T + 11T^{2} \)
13 \( 1 + 3.56iT - 13T^{2} \)
17 \( 1 + 0.895iT - 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 + 7.24iT - 37T^{2} \)
41 \( 1 - 7.84T + 41T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + 2.96iT - 47T^{2} \)
53 \( 1 - 4.78iT - 53T^{2} \)
59 \( 1 + 5.75T + 59T^{2} \)
61 \( 1 - 4.34T + 61T^{2} \)
67 \( 1 + 8.53iT - 67T^{2} \)
71 \( 1 - 5.34T + 71T^{2} \)
73 \( 1 - 9.34iT - 73T^{2} \)
79 \( 1 - 0.741T + 79T^{2} \)
83 \( 1 - 9.21iT - 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 3.45iT - 97T^{2} \)
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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

−9.348486600867366428683051244718, −8.200465526629750689634099701397, −7.44479955199735619827075861915, −7.11268059782040398848743767231, −5.74674411489661892894476754456, −5.34290800776113372583499541035, −3.98588522556983369016279488951, −3.05575010292263130026729677582, −2.28867468010251078215337140843, −0.39336018644049372618428298458, 1.36768163352545050571213633341, 2.49354616142357128127107910198, 3.67903845425486228655264557835, 4.83545241774687972243416787526, 5.36654134870407329839726708256, 6.15243505484315799416401588218, 7.51369402946592631354108319526, 7.87647461999766586241613350238, 8.958319074165759453860853281352, 9.412242954173690778432367795422

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