Properties

Label 2-1200-5.4-c1-0-10
Degree $2$
Conductor $1200$
Sign $0.894 + 0.447i$
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  + i·3-s − 9-s + 4·11-s − 6i·13-s − 6i·17-s − 4·19-s i·27-s + 2·29-s + 8·31-s + 4i·33-s − 2i·37-s + 6·39-s − 6·41-s + 12i·43-s − 8i·47-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s − 1.45i·17-s − 0.917·19-s − 0.192i·27-s + 0.371·29-s + 1.43·31-s + 0.696i·33-s − 0.328i·37-s + 0.960·39-s − 0.937·41-s + 1.82i·43-s − 1.16i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605797357\)
\(L(\frac12)\) \(\approx\) \(1.605797357\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.931262264743306093004710089817, −8.814598430926268027923828107991, −8.277529208404229630937827649933, −7.16580120297517541933710892352, −6.32015188731403822706283273449, −5.35593063783492880148621853191, −4.52770601332007010236933037983, −3.50847780841023239525392880373, −2.56673412766581575731121058677, −0.77146599707915172535366148985, 1.34273256913335249576489420451, 2.27243183320544802701974579879, 3.83277241589066976058645379695, 4.42442984847417417168818826386, 5.87769062104612673901513864072, 6.59658709041959294152260972788, 7.07036372701954498718765370591, 8.489067701080944716754451533930, 8.683498348650710752501376736498, 9.773779295038427916214114622401

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