Properties

Label 2-1200-12.11-c1-0-37
Degree $2$
Conductor $1200$
Sign $-0.866 - 0.5i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 5.19i·7-s − 2.99·9-s − 7·13-s + 5.19i·19-s − 9·21-s + 5.19i·27-s − 1.73i·31-s + 10·37-s + 12.1i·39-s + 1.73i·43-s − 20·49-s + 9·57-s − 61-s + 15.5i·63-s + ⋯
L(s)  = 1  − 0.999i·3-s − 1.96i·7-s − 0.999·9-s − 1.94·13-s + 1.19i·19-s − 1.96·21-s + 0.999i·27-s − 0.311i·31-s + 1.64·37-s + 1.94i·39-s + 0.264i·43-s − 2.85·49-s + 1.19·57-s − 0.128·61-s + 1.96i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.866 - 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6634787623\)
\(L(\frac12)\) \(\approx\) \(0.6634787623\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 + 5.19iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 17.3iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19T + 97T^{2} \)
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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

−9.422550516650503380920603643725, −7.953997253661533954297839297987, −7.65386470864920335632622302971, −6.98435933269876262489796397732, −6.13952396670763948482909527941, −4.92828676989730344467296981644, −4.01857656322000945710657715467, −2.82239737836733782608150438975, −1.53537794213936110850949704367, −0.27172348138090443192381610246, 2.47332103456048729065430526665, 2.80030907518900402726418495515, 4.40213043134028745468022780774, 5.15716719821352489369661815286, 5.71484093423719012499150660632, 6.83269136474204915474153533449, 8.010600507619943124845136573336, 8.835027303624975240967525120916, 9.448549833484785043448176587654, 9.916155051522485848918594032273

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