Properties

Label 2-5400-5.4-c1-0-59
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  − 4i·7-s − 2·11-s − 4i·13-s i·17-s + 5·19-s + 5i·23-s + 8·29-s + 7·31-s − 6i·37-s − 6·41-s + 2i·43-s − 8i·47-s − 9·49-s + 9i·53-s + 4·59-s + ⋯
L(s)  = 1  − 1.51i·7-s − 0.603·11-s − 1.10i·13-s − 0.242i·17-s + 1.14·19-s + 1.04i·23-s + 1.48·29-s + 1.25·31-s − 0.986i·37-s − 0.937·41-s + 0.304i·43-s − 1.16i·47-s − 1.28·49-s + 1.23i·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

\(L(1)\) \(\approx\) \(1.670908421\)
\(L(\frac12)\) \(\approx\) \(1.670908421\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 5iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 + 17iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−7.888779604446513700235520831823, −7.27377081650651491099368540753, −6.71930137280208014404550719121, −5.63898729066692991743344834641, −5.07645861724610998857834536149, −4.22034593240676834943524441166, −3.38695235142391045594797925299, −2.73570912604652435023129315319, −1.28163865748179823440418228252, −0.49394191554712047749289178925, 1.21879466829391126529469352461, 2.44384031563773400378362663838, 2.80163996766485035779918020571, 4.00607476183968351693958705982, 4.98451315301008518332294637764, 5.34373350150100179052004926480, 6.48498415433447582404446566068, 6.64839939934742245942584932295, 7.912628774792344586356678604113, 8.457513604577591734207474731755

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