Properties

Label 2-5400-5.4-c1-0-11
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  − 3i·7-s + 5·11-s + 4i·13-s + 8i·17-s − 2·19-s + 2i·23-s − 6·29-s − 7·31-s + 6i·37-s − 6·41-s − 2i·43-s − 6i·47-s − 2·49-s + 5i·53-s + 4·59-s + ⋯
L(s)  = 1  − 1.13i·7-s + 1.50·11-s + 1.10i·13-s + 1.94i·17-s − 0.458·19-s + 0.417i·23-s − 1.11·29-s − 1.25·31-s + 0.986i·37-s − 0.937·41-s − 0.304i·43-s − 0.875i·47-s − 0.285·49-s + 0.686i·53-s + 0.520·59-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

\(L(1)\) \(\approx\) \(1.076106431\)
\(L(\frac12)\) \(\approx\) \(1.076106431\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−8.615495351766044192084741097264, −7.50564367765200221301608759911, −6.98979661228171531634185494352, −6.36206863660346596386242591381, −5.68238009373076973321178143258, −4.41420557126733876977935607197, −3.98694790468124806575689375957, −3.47778947356838967625892409343, −1.80504140097477334791153832179, −1.41053815764737135214798380682, 0.27195242167192809556675305017, 1.60253438084587106743806142977, 2.58043159623514795103542932919, 3.32617298361677094913514036588, 4.23797212329524637497636238290, 5.21520433357114184351626430047, 5.67854238736469912943297810963, 6.50783527736378785509248610138, 7.21787666304516195158655795546, 7.920959637946580340140411278262

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