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Properties

Label 2-1200-5.2-c2-0-3
Degree $2$
Conductor $1200$
Sign $-0.229 - 0.973i$
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  + (−1.22 + 1.22i)3-s + (−8.89 − 8.89i)7-s − 2.99i·9-s − 5.79·11-s + (6.79 − 6.79i)13-s + (−6.10 − 6.10i)17-s − 6.20i·19-s + 21.7·21-s + (−18.6 + 18.6i)23-s + (3.67 + 3.67i)27-s − 6.20i·29-s + 0.404·31-s + (7.10 − 7.10i)33-s + (27 + 27i)37-s + 16.6i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−1.27 − 1.27i)7-s − 0.333i·9-s − 0.527·11-s + (0.522 − 0.522i)13-s + (−0.358 − 0.358i)17-s − 0.326i·19-s + 1.03·21-s + (−0.812 + 0.812i)23-s + (0.136 + 0.136i)27-s − 0.213i·29-s + 0.0130·31-s + (0.215 − 0.215i)33-s + (0.729 + 0.729i)37-s + 0.426i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4678874545\)
\(L(\frac12)\) \(\approx\) \(0.4678874545\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (8.89 + 8.89i)T + 49iT^{2} \)
11 \( 1 + 5.79T + 121T^{2} \)
13 \( 1 + (-6.79 + 6.79i)T - 169iT^{2} \)
17 \( 1 + (6.10 + 6.10i)T + 289iT^{2} \)
19 \( 1 + 6.20iT - 361T^{2} \)
23 \( 1 + (18.6 - 18.6i)T - 529iT^{2} \)
29 \( 1 + 6.20iT - 841T^{2} \)
31 \( 1 - 0.404T + 961T^{2} \)
37 \( 1 + (-27 - 27i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.79T + 1.68e3T^{2} \)
43 \( 1 + (-36.4 + 36.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-38.6 - 38.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (69.0 - 69.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 + 63.1T + 3.72e3T^{2} \)
67 \( 1 + (-40.0 - 40.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 25.7T + 5.04e3T^{2} \)
73 \( 1 + (-56.7 + 56.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 139. iT - 6.24e3T^{2} \)
83 \( 1 + (-13.7 + 13.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 58.6iT - 7.92e3T^{2} \)
97 \( 1 + (-15.9 - 15.9i)T + 9.40e3iT^{2} \)
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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

−9.775372796753835780709115478299, −9.320330197204647008640191784823, −8.023805473769997860744260914380, −7.27379383151079173714093591699, −6.40081315732320499601694244560, −5.69348605993407217067187716764, −4.49908606880532223348663869202, −3.72406963077434138017780801432, −2.81245487252358025399883340755, −0.916491925422925666755536836958, 0.18174165377111704725271937790, 1.96272098731485425765894182332, 2.86748045631290263245407436353, 4.04643032586110634485668800596, 5.29655567961762311232700818379, 6.17739106006608488847600468770, 6.47104725924836479554086992557, 7.68763395596647453824532070655, 8.589590952522753906988499763417, 9.273296936686174299781146208809

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