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Properties

Label 2-1200-12.11-c1-0-10
Degree $2$
Conductor $1200$
Sign $-0.866 - 0.5i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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  + 1.73i·3-s + 5.19i·7-s − 2.99·9-s + 7·13-s + 5.19i·19-s − 9·21-s − 5.19i·27-s − 1.73i·31-s − 10·37-s + 12.1i·39-s − 1.73i·43-s − 20·49-s − 9·57-s − 61-s − 15.5i·63-s + ⋯
L(s)  = 1  + 0.999i·3-s + 1.96i·7-s − 0.999·9-s + 1.94·13-s + 1.19i·19-s − 1.96·21-s − 0.999i·27-s − 0.311i·31-s − 1.64·37-s + 1.94i·39-s − 0.264i·43-s − 2.85·49-s − 1.19·57-s − 0.128·61-s − 1.96i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.866 - 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.483583614\)
\(L(\frac12)\) \(\approx\) \(1.483583614\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 5.19iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 17.3iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−10.05769130654097618758111806067, −9.069612726620187540717790281186, −8.701699375239207520995523548335, −8.054530776839527414236421257152, −6.34559182124706036108067872100, −5.80713185390175186171157190897, −5.15650859983305851262442063608, −3.87370064190643972232271268996, −3.12101046085674015410215286326, −1.87657160997623496945792874564, 0.66806757366720855191734551195, 1.57078476498766989933462670894, 3.22351797994124113868420109577, 3.99335382965830650528329180599, 5.19039680580781884217894083783, 6.47864904822841063372522390551, 6.80832493091351651812563599795, 7.70766500544211890785719330700, 8.402956796952966865129499966223, 9.270005683140784836591780933170

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