Properties

Label 2-1200-12.11-c1-0-14
Degree $2$
Conductor $1200$
Sign $0.5 - 0.866i$
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 + 3.46i·7-s + 2.99·9-s + 3.46·11-s − 4·13-s + 6i·17-s + 3.46i·19-s + 5.99i·21-s − 3.46·23-s + 5.19·27-s − 6i·29-s − 3.46i·31-s + 5.99·33-s + 4·37-s − 6.92·39-s + ⋯
L(s)  = 1  + 1.00·3-s + 1.30i·7-s + 0.999·9-s + 1.04·11-s − 1.10·13-s + 1.45i·17-s + 0.794i·19-s + 1.30i·21-s − 0.722·23-s + 1.00·27-s − 1.11i·29-s − 0.622i·31-s + 1.04·33-s + 0.657·37-s − 1.10·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.5 - 0.866i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326890053\)
\(L(\frac12)\) \(\approx\) \(2.326890053\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 6.92iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.617174337322812066821998858002, −9.154497285542077631355645209698, −8.207093751741031091844427746137, −7.76431221982458566106021140880, −6.46808189778991609006335920607, −5.84430080541105412271687470428, −4.51559807789055526208734608411, −3.70721354746632601359932475889, −2.51901527819013492998931905679, −1.74616850029122072339844568918, 0.927406560044210912211595088998, 2.34015288593240148834715663035, 3.42830021632819486182643434622, 4.27926499727315289089739939820, 5.06278832377350930361764924146, 6.78820023125593017989067828946, 7.11360322846134384945366638025, 7.85498510121115723901522351805, 8.972931571660743922204958213263, 9.509454266858874177720613326992

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