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Properties

Label 2-7200-12.11-c1-0-29
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  − 2.64i·7-s − 3.00·11-s − 0.640·13-s + 0.685i·17-s + 5.28i·19-s − 2.27·23-s + 8.15i·29-s − 2.96i·31-s − 1.60·37-s − 7.42i·41-s − 11.2i·43-s + 4.19·47-s + 0.0302·49-s + 9.60i·53-s + 7.20·59-s + ⋯
L(s)  = 1  − 0.997i·7-s − 0.906·11-s − 0.177·13-s + 0.166i·17-s + 1.21i·19-s − 0.474·23-s + 1.51i·29-s − 0.533i·31-s − 0.264·37-s − 1.15i·41-s − 1.71i·43-s + 0.612·47-s + 0.00432·49-s + 1.32i·53-s + 0.937·59-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

\(L(1)\) \(\approx\) \(1.566502239\)
\(L(\frac12)\) \(\approx\) \(1.566502239\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 + 3.00T + 11T^{2} \)
13 \( 1 + 0.640T + 13T^{2} \)
17 \( 1 - 0.685iT - 17T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + 2.96iT - 31T^{2} \)
37 \( 1 + 1.60T + 37T^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 - 4.19T + 47T^{2} \)
53 \( 1 - 9.60iT - 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 8.49iT - 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−7.83101423901312264928410448840, −7.24384991404295751827053923005, −6.68910317549539635808682287497, −5.57277178546273844940531488182, −5.29757316328270114315782749668, −4.06029729313640276674004900092, −3.78543702031784166915758103309, −2.66804529562678950694134528530, −1.76104898307774120224567532467, −0.63338393126624327397536677045, 0.60400937756915185220573929959, 2.09840518363306605307013145881, 2.60183805498138299835630346096, 3.43598867930708619469108581936, 4.60758554059092183221058661698, 5.04316295257658330084623264317, 5.87376902008542073979023023445, 6.44342514634435464356696935372, 7.31173801154771809620676510099, 8.027818045958237560632737253446

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