Properties

Label 2-5400-5.4-c1-0-22
Degree $2$
Conductor $5400$
Sign $0.447 - 0.894i$
Analytic cond. $43.1192$
Root an. cond. $6.56652$
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  + 2i·7-s − 4·11-s + 2i·13-s − 5i·17-s + 5·19-s + i·23-s − 2·29-s + 7·31-s − 6i·37-s − 4i·43-s − 4i·47-s + 3·49-s + 9i·53-s + 14·59-s − 11·61-s + ⋯
L(s)  = 1  + 0.755i·7-s − 1.20·11-s + 0.554i·13-s − 1.21i·17-s + 1.14·19-s + 0.208i·23-s − 0.371·29-s + 1.25·31-s − 0.986i·37-s − 0.609i·43-s − 0.583i·47-s + 0.428·49-s + 1.23i·53-s + 1.82·59-s − 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5400\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(43.1192\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5400} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.572430336\)
\(L(\frac12)\) \(\approx\) \(1.572430336\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 14T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−8.345270183361947044599939969315, −7.44615503305948131248514102896, −7.10789473179686956302803286314, −5.97811384516288082467007100774, −5.38505375582839570961371897164, −4.84453364938573235203482031350, −3.80465457221140442853936743674, −2.75859437368975164278226958323, −2.32446047100163260834764846634, −0.902818176387185737754311669214, 0.51320670619519088892062977641, 1.62541473600245785047055149123, 2.81095020440496331154088398882, 3.45089141863871636893810066595, 4.42867012941824790532438985113, 5.11400302517237359093259604316, 5.88320244115332881198355048646, 6.62069475559202915648586693719, 7.51341788346161165351386086067, 7.975715896203517819245614427972

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