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Properties

Label 2-7200-8.5-c1-0-32
Degree $2$
Conductor $7200$
Sign $-0.187 - 0.982i$
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  + 0.941·7-s + 4.49i·11-s + 5.55i·13-s + 7.55·17-s − 1.05i·19-s + 1.05·23-s + 2i·29-s − 3.55·31-s + 7.43i·37-s + 3.88·41-s − 1.88i·43-s + 10.0·47-s − 6.11·49-s + 2i·53-s − 8.49i·59-s + ⋯
L(s)  = 1  + 0.355·7-s + 1.35i·11-s + 1.54i·13-s + 1.83·17-s − 0.242i·19-s + 0.220·23-s + 0.371i·29-s − 0.638·31-s + 1.22i·37-s + 0.606·41-s − 0.287i·43-s + 1.46·47-s − 0.873·49-s + 0.274i·53-s − 1.10i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (3601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.991316042\)
\(L(\frac12)\) \(\approx\) \(1.991316042\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.941T + 7T^{2} \)
11 \( 1 - 4.49iT - 11T^{2} \)
13 \( 1 - 5.55iT - 13T^{2} \)
17 \( 1 - 7.55T + 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 3.55T + 31T^{2} \)
37 \( 1 - 7.43iT - 37T^{2} \)
41 \( 1 - 3.88T + 41T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 8.49iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 5.88iT - 83T^{2} \)
89 \( 1 - 4.11T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−7.974497174132713108641255425026, −7.36259162264428942192424444428, −6.90217685866454480215662604450, −6.03300250475470378387140173264, −5.19325595836720581590605929200, −4.57582059078802983905055309716, −3.90611107150007893189547401643, −2.92879981315349072503112350433, −1.92243190110837109144139987108, −1.25937066239989257878753328726, 0.52284931552214280159840540939, 1.30627936697431948076159889722, 2.66884953527637305723649818248, 3.30072697707591921596681194477, 3.95429138477709466938564095500, 5.12192511257096068279738529100, 5.75864566789684162520438147809, 5.95287108172487595789873623781, 7.29196051502541241427005762135, 7.74804911385026751730194903121

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