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Properties

Label 2-7200-40.29-c1-0-19
Degree $2$
Conductor $7200$
Sign $-0.844 - 0.536i$
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  + 3.62i·7-s + 6.20i·11-s − 0.578·13-s − 1.42i·17-s − 5.62i·19-s + 5.62i·23-s + 2i·29-s + 2.57·31-s + 7.83·37-s − 5.25·41-s + 7.25·43-s + 6.78i·47-s − 6.15·49-s + 2·53-s + 2.20i·59-s + ⋯
L(s)  = 1  + 1.37i·7-s + 1.87i·11-s − 0.160·13-s − 0.344i·17-s − 1.29i·19-s + 1.17i·23-s + 0.371i·29-s + 0.463·31-s + 1.28·37-s − 0.820·41-s + 1.10·43-s + 0.989i·47-s − 0.879·49-s + 0.274·53-s + 0.287i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.844 - 0.536i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.844 - 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.523481000\)
\(L(\frac12)\) \(\approx\) \(1.523481000\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.62iT - 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 + 0.578T + 13T^{2} \)
17 \( 1 + 1.42iT - 17T^{2} \)
19 \( 1 + 5.62iT - 19T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 + 5.25T + 41T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 - 6.78iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 8.41T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 - 3.25T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 4.84iT - 97T^{2} \)
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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

−8.199413932743994849167248499295, −7.39188549775802867366049384409, −6.95418697796436433879837243181, −6.07748433099617882946295337223, −5.29124077967707300178471076030, −4.79263593322519760551646207668, −4.01351455437297958205452461789, −2.69588400804489880803599262782, −2.41854975370956662093350388954, −1.32171105544503658245726529245, 0.40628038905798949174375705398, 1.12840561205083510891657294757, 2.39720889123347830048113779802, 3.46300036666273292565506996775, 3.86079646097466538375952454262, 4.69496493383207899574357893107, 5.66369548468206495362389940555, 6.28232702273336091218632102952, 6.83555347566210259835135259815, 7.927232746786282507898373629992

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