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Properties

Label 2-1200-20.19-c4-0-48
Degree $2$
Conductor $1200$
Sign $0.998 + 0.0599i$
Analytic cond. $124.043$
Root an. cond. $11.1375$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19·3-s + 76.2·7-s + 27·9-s − 20.7i·11-s − 182i·13-s + 246i·17-s + 117. i·19-s + 396·21-s + 748.·23-s + 140.·27-s − 78·29-s + 1.47e3i·31-s − 108i·33-s − 530i·37-s − 945. i·39-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.55·7-s + 0.333·9-s − 0.171i·11-s − 1.07i·13-s + 0.851i·17-s + 0.326i·19-s + 0.897·21-s + 1.41·23-s + 0.192·27-s − 0.0927·29-s + 1.53i·31-s − 0.0991i·33-s − 0.387i·37-s − 0.621i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0599i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 + 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.998 + 0.0599i$
Analytic conductor: \(124.043\)
Root analytic conductor: \(11.1375\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :2),\ 0.998 + 0.0599i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.843053060\)
\(L(\frac12)\) \(\approx\) \(3.843053060\)
\(L(3)\) 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 \( 1 - 5.19T \)
5 \( 1 \)
good7 \( 1 - 76.2T + 2.40e3T^{2} \)
11 \( 1 + 20.7iT - 1.46e4T^{2} \)
13 \( 1 + 182iT - 2.85e4T^{2} \)
17 \( 1 - 246iT - 8.35e4T^{2} \)
19 \( 1 - 117. iT - 1.30e5T^{2} \)
23 \( 1 - 748.T + 2.79e5T^{2} \)
29 \( 1 + 78T + 7.07e5T^{2} \)
31 \( 1 - 1.47e3iT - 9.23e5T^{2} \)
37 \( 1 + 530iT - 1.87e6T^{2} \)
41 \( 1 + 918T + 2.82e6T^{2} \)
43 \( 1 - 852.T + 3.41e6T^{2} \)
47 \( 1 - 3.78e3T + 4.87e6T^{2} \)
53 \( 1 + 4.62e3iT - 7.89e6T^{2} \)
59 \( 1 - 228. iT - 1.21e7T^{2} \)
61 \( 1 - 1.34e3T + 1.38e7T^{2} \)
67 \( 1 - 1.08e3T + 2.01e7T^{2} \)
71 \( 1 + 1.82e3iT - 2.54e7T^{2} \)
73 \( 1 + 926iT - 2.83e7T^{2} \)
79 \( 1 - 4.39e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.19e4T + 4.74e7T^{2} \)
89 \( 1 + 1.15e4T + 6.27e7T^{2} \)
97 \( 1 - 1.31e4iT - 8.85e7T^{2} \)
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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

−8.857049044648111578260714362635, −8.431002121128509405115634650367, −7.72118377778342270636948899010, −6.92438887601172147425612573280, −5.60038515189484643115630878303, −4.97980672455602196511747530495, −3.93540556875925202670630829973, −2.93814651149690795310535718922, −1.80565579456560816887783470374, −0.930389056317729983468221288926, 0.927836178575106615151128281356, 1.92416442793797120360348361446, 2.79636972559070983209809847262, 4.22289183795621717788200560981, 4.71012125441720946158322138406, 5.72856940885095564316362845824, 7.09294731061970624039647953669, 7.47080691103830052112735291619, 8.515571400100647203767504676489, 9.043948921369278191467161277302

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