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Properties

Label 2-1200-60.59-c1-0-32
Degree $2$
Conductor $1200$
Sign $-0.0599 + 0.998i$
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  + 1.73·3-s − 5.19·7-s + 2.99·9-s − 7i·13-s − 5.19i·19-s − 9·21-s + 5.19·27-s − 1.73i·31-s − 10i·37-s − 12.1i·39-s − 1.73·43-s + 20·49-s − 9i·57-s − 61-s − 15.5·63-s + ⋯
L(s)  = 1  + 1.00·3-s − 1.96·7-s + 0.999·9-s − 1.94i·13-s − 1.19i·19-s − 1.96·21-s + 1.00·27-s − 0.311i·31-s − 1.64i·37-s − 1.94i·39-s − 0.264·43-s + 2.85·49-s − 1.19i·57-s − 0.128·61-s − 1.96·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0599 + 0.998i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.0599 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.483583614\)
\(L(\frac12)\) \(\approx\) \(1.483583614\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 + 5.19T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7iT - 13T^{2} \)
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 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.73T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 17.3iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 19iT - 97T^{2} \)
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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

−9.428233673211750949047553252363, −8.922581265018907633205072868441, −7.87007210639657550789323976059, −7.15893934530357559403133607722, −6.31518126145457791205752855917, −5.35585424263450399598588658806, −3.96508712736863941057538495972, −3.13719144071411459267368978091, −2.57865312731521128586671110581, −0.54396303043683315856031446211, 1.69190996967345702015385543830, 2.91343342706374562342221559431, 3.67017533112789168414992746728, 4.48921633860758638676170169738, 6.09219501933549044042707567337, 6.69477016761271122260977606262, 7.39017096557644175614159673508, 8.570845947295107052848144113078, 9.174075088356045731340388198781, 9.843954975854656828477445957845

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