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Properties

Label 2-1200-240.83-c0-0-1
Degree $2$
Conductor $1200$
Sign $-0.584 + 0.811i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − i·3-s + 4-s + i·6-s − 8-s − 9-s − i·12-s + 16-s + (−1 − i)17-s + 18-s + (−1 − i)19-s + (1 − i)23-s + i·24-s + i·27-s − 2i·31-s − 32-s + ⋯
L(s)  = 1  − 2-s − i·3-s + 4-s + i·6-s − 8-s − 9-s − i·12-s + 16-s + (−1 − i)17-s + 18-s + (−1 − i)19-s + (1 − i)23-s + i·24-s + i·27-s − 2i·31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.584 + 0.811i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ -0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5317437567\)
\(L(\frac12)\) \(\approx\) \(0.5317437567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.251065716201073465975765640428, −8.958819518630306750735421541738, −8.016274656817931161894764196798, −7.22514497544766569979336867921, −6.63894696983814711104810248890, −5.86256817537407808931962074365, −4.53725946997020161386281139161, −2.80805086292252744111457222456, −2.19716204253244696663132773602, −0.61960754735892546299978320216, 1.78190024764347450669228785225, 3.08946936508653771795112343695, 4.02476316341054175305399621095, 5.23600874176043722946185523744, 6.15953677823768216065772613776, 6.97856812361007893798053660801, 8.162484288652240273544613677108, 8.708829157383072054403462914759, 9.348672313368954692765408602787, 10.33627604212210333964710070170

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