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Properties

Label 2-1200-240.203-c0-0-0
Degree $2$
Conductor $1200$
Sign $-0.160 - 0.987i$
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  + i·2-s − 3-s − 4-s − i·6-s − i·8-s + 9-s + 12-s + 16-s + (1 + i)17-s + i·18-s + (1 − i)19-s + (−1 + i)23-s + i·24-s − 27-s + 2i·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s − i·6-s − i·8-s + 9-s + 12-s + 16-s + (1 + i)17-s + i·18-s + (1 − i)19-s + (−1 + i)23-s + i·24-s − 27-s + 2i·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.160 - 0.987i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ -0.160 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6919550510\)
\(L(\frac12)\) \(\approx\) \(0.6919550510\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
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 \( 1 - 2T + T^{2} \)
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

−10.12738885703313631899197824065, −9.348439253969716103499452070385, −8.397423941846263771717808358874, −7.42159083927956111643318141993, −6.92886345947696367569402693165, −5.80669513636148049916062697368, −5.44858057555189492055433108603, −4.42135424615880239583689128040, −3.44437852206595666559099523709, −1.25506498967479330315190931877, 0.861272831468017989123779469564, 2.25868528192296967182604056390, 3.60230965685319239762687001590, 4.45815436208753662970914316506, 5.44554118987709071019299583272, 6.02310815220676841352582560109, 7.39631564949780711285228755408, 8.059974699900014531791475717678, 9.342109064604868001304213804416, 9.934631146403040635609845463874

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