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Properties

Label 2-7200-5.4-c1-0-41
Degree $2$
Conductor $7200$
Sign $0.894 + 0.447i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  + 2.82i·7-s − 5.65·11-s + 2i·13-s − 2i·17-s − 2.82i·23-s + 6·29-s − 5.65·31-s − 10i·37-s − 2·41-s − 8.48i·43-s − 2.82i·47-s − 1.00·49-s + 6i·53-s − 11.3·59-s − 2·61-s + ⋯
L(s)  = 1  + 1.06i·7-s − 1.70·11-s + 0.554i·13-s − 0.485i·17-s − 0.589i·23-s + 1.11·29-s − 1.01·31-s − 1.64i·37-s − 0.312·41-s − 1.29i·43-s − 0.412i·47-s − 0.142·49-s + 0.824i·53-s − 1.47·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266166508\)
\(L(\frac12)\) \(\approx\) \(1.266166508\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2.82iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−7.82094732656024622994843459021, −7.29380197993992485935342729542, −6.42764372962017001071778309508, −5.61633048900528609860934201220, −5.19218545621568710898378051732, −4.42741845741385986054532101570, −3.35003215205336487942276309473, −2.50122322779165823029852736069, −2.04335727876463045059910511894, −0.41996540591292181751682563827, 0.72303286314502468889046315025, 1.82703938522790392996229776317, 2.95501577763324480433586879451, 3.45821674675477819568040917461, 4.57668138792436507615124268942, 5.01141070697186690976656219957, 5.88511205878476361218286964804, 6.61005727794237701068526685585, 7.45506270835760872358901200625, 7.921925426099951868678946160967

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