Properties

Label 2-7200-5.4-c1-0-25
Degree $2$
Conductor $7200$
Sign $0.447 - 0.894i$
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  − 2i·7-s + 5·11-s + 5i·17-s + 5·19-s + 6i·23-s + 4·29-s − 10·31-s + 10i·37-s − 5·41-s − 4i·43-s + 8i·47-s + 3·49-s + 10i·53-s − 10·61-s + 3i·67-s + ⋯
L(s)  = 1  − 0.755i·7-s + 1.50·11-s + 1.21i·17-s + 1.14·19-s + 1.25i·23-s + 0.742·29-s − 1.79·31-s + 1.64i·37-s − 0.780·41-s − 0.609i·43-s + 1.16i·47-s + 0.428·49-s + 1.37i·53-s − 1.28·61-s + 0.366i·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 7200,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976337789\)
\(L(\frac12)\) \(\approx\) \(1.976337789\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−7.893463127453781428065205398208, −7.37531108897141827099665843771, −6.66817480820214026043513226131, −6.04218976959245953152140623554, −5.25443699986158129421127384433, −4.31334600879724889704186804431, −3.72275047826457630248821166216, −3.11649486796155157017523672696, −1.61450444403337656299925740886, −1.18551866908015971922614456941, 0.51092269099572287910079071898, 1.65196290367657739767103631887, 2.55687230888753316463160803997, 3.42169017106700722991706893231, 4.15555517783461570217754808179, 5.11248210463901629885957202798, 5.58733519521942710924459766528, 6.55556994742163320850891697567, 6.97103644968742980111144529460, 7.75935638861024541399912663224

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