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Properties

Label 2-1200-3.2-c2-0-50
Degree $2$
Conductor $1200$
Sign $0.666 + 0.745i$
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 + 2.23i)3-s − 8·7-s + (−1.00 + 8.94i)9-s − 8.94i·11-s + 12·13-s − 31.3i·17-s − 6·19-s + (−16 − 17.8i)21-s + 4.47i·23-s + (−22.0 + 15.6i)27-s − 26.8i·29-s − 34·31-s + (20.0 − 17.8i)33-s + 44·37-s + (24 + 26.8i)39-s + ⋯
L(s)  = 1  + (0.666 + 0.745i)3-s − 1.14·7-s + (−0.111 + 0.993i)9-s − 0.813i·11-s + 0.923·13-s − 1.84i·17-s − 0.315·19-s + (−0.761 − 0.851i)21-s + 0.194i·23-s + (−0.814 + 0.579i)27-s − 0.925i·29-s − 1.09·31-s + (0.606 − 0.542i)33-s + 1.18·37-s + (0.615 + 0.688i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.666 + 0.745i$
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: \((2,\ 1200,\ (\ :1),\ 0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.672305201\)
\(L(\frac12)\) \(\approx\) \(1.672305201\)
\(L(2)\) 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 \( 1 + (-2 - 2.23i)T \)
5 \( 1 \)
good7 \( 1 + 8T + 49T^{2} \)
11 \( 1 + 8.94iT - 121T^{2} \)
13 \( 1 - 12T + 169T^{2} \)
17 \( 1 + 31.3iT - 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 - 4.47iT - 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 + 34T + 961T^{2} \)
37 \( 1 - 44T + 1.36e3T^{2} \)
41 \( 1 + 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 28T + 1.84e3T^{2} \)
47 \( 1 + 4.47iT - 2.20e3T^{2} \)
53 \( 1 + 40.2iT - 2.80e3T^{2} \)
59 \( 1 + 98.3iT - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 - 92T + 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 56T + 5.32e3T^{2} \)
79 \( 1 + 78T + 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 + 17.8iT - 7.92e3T^{2} \)
97 \( 1 + 32T + 9.40e3T^{2} \)
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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

−9.491362590206890531935289695882, −8.808050031617336576180631260548, −7.975294939684548847343027349580, −6.99233446697566678145426715346, −6.03198557976057291601956420863, −5.16594075755785682266867686534, −3.95831363824835224703924530203, −3.29591769873658924393318082142, −2.40537621455211943689836759981, −0.47680570151828622420258285760, 1.23302541003468110864519699742, 2.33125157415396602263468718351, 3.46950251720632822980462670553, 4.12952749495053036363055055534, 5.80737177424087145750418558197, 6.41505152691108645914084264257, 7.14525525634516896475142155311, 8.080436648943174777708516971532, 8.810979434028453830621339879756, 9.517256147066829449582634919801

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