Properties

Label 2-1200-15.14-c2-0-10
Degree $2$
Conductor $1200$
Sign $-0.807 - 0.590i$
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  + (−2.95 − 0.5i)3-s + 8i·7-s + (8.5 + 2.95i)9-s + 17.7i·11-s + 2i·13-s + 17.7·17-s + 11·19-s + (4 − 23.6i)21-s − 35.4·23-s + (−23.6 − 13i)27-s + 35.4i·29-s + 46·31-s + (8.87 − 52.5i)33-s + 16i·37-s + (1 − 5.91i)39-s + ⋯
L(s)  = 1  + (−0.986 − 0.166i)3-s + 1.14i·7-s + (0.944 + 0.328i)9-s + 1.61i·11-s + 0.153i·13-s + 1.04·17-s + 0.578·19-s + (0.190 − 1.12i)21-s − 1.54·23-s + (−0.876 − 0.481i)27-s + 1.22i·29-s + 1.48·31-s + (0.268 − 1.59i)33-s + 0.432i·37-s + (0.0256 − 0.151i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.807 - 0.590i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.807 - 0.590i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9435559476\)
\(L(\frac12)\) \(\approx\) \(0.9435559476\)
\(L(2)\) 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 \( 1 + (2.95 + 0.5i)T \)
5 \( 1 \)
good7 \( 1 - 8iT - 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 - 2iT - 169T^{2} \)
17 \( 1 - 17.7T + 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + 35.4T + 529T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 - 46T + 961T^{2} \)
37 \( 1 - 16iT - 1.36e3T^{2} \)
41 \( 1 + 53.2iT - 1.68e3T^{2} \)
43 \( 1 + 62iT - 1.84e3T^{2} \)
47 \( 1 + 35.4T + 2.20e3T^{2} \)
53 \( 1 + 35.4T + 2.80e3T^{2} \)
59 \( 1 - 70.9iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 - 113iT - 4.48e3T^{2} \)
71 \( 1 + 106. iT - 5.04e3T^{2} \)
73 \( 1 - 101iT - 5.32e3T^{2} \)
79 \( 1 - 68T + 6.24e3T^{2} \)
83 \( 1 - 17.7T + 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 - 22iT - 9.40e3T^{2} \)
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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

−10.00067276051882465218870789641, −9.251877593983138171155270187284, −8.124593743144683118818875373534, −7.30385181973812917947639822269, −6.51280482539555639546734414919, −5.56633841750845384883456934740, −5.02368519325453347194959849554, −3.96113236563399997059489270789, −2.43601108656783332642764654077, −1.43245983898348329395015517444, 0.36663172597538225547862270408, 1.22430071601473021767067985341, 3.17895047070393791056631547032, 4.02759186349882828533524722298, 4.96733628606506432308480739509, 6.06086552444017296486121887015, 6.38188353578158092494323675043, 7.77443217649436340053750940842, 8.056822830615782512057745565837, 9.637959768943008448873969058938

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