Properties

Label 2-7200-12.11-c1-0-44
Degree $2$
Conductor $7200$
Sign $0.985 + 0.169i$
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  − 0.864i·7-s + 3.90·11-s + 1.13·13-s − 3.71i·17-s + 1.72i·19-s + 9.03·23-s − 1.26i·29-s + 3.25i·31-s + 6.38·37-s + 6.39i·41-s + 4.77i·43-s − 4.59·47-s + 6.25·49-s + 8.98i·53-s − 8.50·59-s + ⋯
L(s)  = 1  − 0.326i·7-s + 1.17·11-s + 0.314·13-s − 0.900i·17-s + 0.396i·19-s + 1.88·23-s − 0.235i·29-s + 0.584i·31-s + 1.05·37-s + 0.998i·41-s + 0.728i·43-s − 0.670·47-s + 0.893·49-s + 1.23i·53-s − 1.10·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.404974473\)
\(L(\frac12)\) \(\approx\) \(2.404974473\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.864iT - 7T^{2} \)
11 \( 1 - 3.90T + 11T^{2} \)
13 \( 1 - 1.13T + 13T^{2} \)
17 \( 1 + 3.71iT - 17T^{2} \)
19 \( 1 - 1.72iT - 19T^{2} \)
23 \( 1 - 9.03T + 23T^{2} \)
29 \( 1 + 1.26iT - 29T^{2} \)
31 \( 1 - 3.25iT - 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 - 6.39iT - 41T^{2} \)
43 \( 1 - 4.77iT - 43T^{2} \)
47 \( 1 + 4.59T + 47T^{2} \)
53 \( 1 - 8.98iT - 53T^{2} \)
59 \( 1 + 8.50T + 59T^{2} \)
61 \( 1 + 9.04T + 61T^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 + 8.10T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 - 8.10T + 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−7.72533288485868182370066040340, −7.28753044016099497720230623724, −6.43244906370162764964953010229, −6.02362371191419407025157172513, −4.83999451260727822828971858013, −4.49493346058789460247944738836, −3.41765140174528458986790871212, −2.89013679605357945537880425349, −1.57083226138984972425525395903, −0.834845879478077752826940407930, 0.844126888355816544380332538854, 1.75483872873166231458552793326, 2.77158899330538585663484540649, 3.63462525427645267274563723763, 4.28664313470383433673666654965, 5.15651044459697362125410673445, 5.89395794322934655258536028212, 6.59095402148912456140625929484, 7.11972581898316141305677458283, 7.982978687693812555593599360259

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