Properties

Label 2-1200-3.2-c2-0-72
Degree $2$
Conductor $1200$
Sign $-0.166 - 0.986i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 2.95i)3-s − 8·7-s + (−8.5 + 2.95i)9-s − 17.7i·11-s + 2·13-s − 17.7i·17-s − 11·19-s + (4 + 23.6i)21-s − 35.4i·23-s + (13 + 23.6i)27-s + 35.4i·29-s + 46·31-s + (−52.5 + 8.87i)33-s − 16·37-s + (−1 − 5.91i)39-s + ⋯
L(s)  = 1  + (−0.166 − 0.986i)3-s − 1.14·7-s + (−0.944 + 0.328i)9-s − 1.61i·11-s + 0.153·13-s − 1.04i·17-s − 0.578·19-s + (0.190 + 1.12i)21-s − 1.54i·23-s + (0.481 + 0.876i)27-s + 1.22i·29-s + 1.48·31-s + (−1.59 + 0.268i)33-s − 0.432·37-s + (−0.0256 − 0.151i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.166 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1161656068\)
\(L(\frac12)\) \(\approx\) \(0.1161656068\)
\(L(2)\) 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 \( 1 + (0.5 + 2.95i)T \)
5 \( 1 \)
good7 \( 1 + 8T + 49T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 - 2T + 169T^{2} \)
17 \( 1 + 17.7iT - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + 35.4iT - 529T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 - 46T + 961T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 - 53.2iT - 1.68e3T^{2} \)
43 \( 1 + 62T + 1.84e3T^{2} \)
47 \( 1 - 35.4iT - 2.20e3T^{2} \)
53 \( 1 + 35.4iT - 2.80e3T^{2} \)
59 \( 1 - 70.9iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 + 113T + 4.48e3T^{2} \)
71 \( 1 - 106. iT - 5.04e3T^{2} \)
73 \( 1 - 101T + 5.32e3T^{2} \)
79 \( 1 + 68T + 6.24e3T^{2} \)
83 \( 1 - 17.7iT - 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 + 22T + 9.40e3T^{2} \)
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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

−8.623996292689705359198859012905, −8.380367797583675783034876398206, −7.08091211860299870299219031553, −6.45731926755107943914860482143, −5.91902057545992538086305392346, −4.76513552661245526947322344817, −3.22668108765024435143525169275, −2.72367440186264164802364577018, −1.03259328635969993888585228875, −0.03991944064517858759358523238, 1.99474838729571371493396398399, 3.29830378084458742007142922255, 4.06346019821561281276721173990, 4.92836355218241569973819037582, 6.00103506612536207990726141795, 6.64354968376629565416730529172, 7.72742869616064280849844850322, 8.731324061530895798557912495062, 9.605289258876036148875300587766, 10.01274096032803513591751294553

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