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Properties

Label 2-1200-5.2-c2-0-26
Degree $2$
Conductor $1200$
Sign $0.229 + 0.973i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (0.898 + 0.898i)7-s − 2.99i·9-s + 13.7·11-s + (−12.7 + 12.7i)13-s + (−15.8 − 15.8i)17-s − 25.7i·19-s + 2.20·21-s + (10.6 − 10.6i)23-s + (−3.67 − 3.67i)27-s − 25.7i·29-s + 39.5·31-s + (16.8 − 16.8i)33-s + (27 + 27i)37-s + 31.3i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.128 + 0.128i)7-s − 0.333i·9-s + 1.25·11-s + (−0.984 + 0.984i)13-s + (−0.935 − 0.935i)17-s − 1.35i·19-s + 0.104·21-s + (0.465 − 0.465i)23-s + (−0.136 − 0.136i)27-s − 0.889i·29-s + 1.27·31-s + (0.512 − 0.512i)33-s + (0.729 + 0.729i)37-s + 0.803i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.108420946\)
\(L(\frac12)\) \(\approx\) \(2.108420946\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-0.898 - 0.898i)T + 49iT^{2} \)
11 \( 1 - 13.7T + 121T^{2} \)
13 \( 1 + (12.7 - 12.7i)T - 169iT^{2} \)
17 \( 1 + (15.8 + 15.8i)T + 289iT^{2} \)
19 \( 1 + 25.7iT - 361T^{2} \)
23 \( 1 + (-10.6 + 10.6i)T - 529iT^{2} \)
29 \( 1 + 25.7iT - 841T^{2} \)
31 \( 1 - 39.5T + 961T^{2} \)
37 \( 1 + (-27 - 27i)T + 1.36e3iT^{2} \)
41 \( 1 - 17.7T + 1.68e3T^{2} \)
43 \( 1 + (12.4 - 12.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.30 - 9.30i)T + 2.20e3iT^{2} \)
53 \( 1 + (-19.0 + 19.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 - 15.1T + 3.72e3T^{2} \)
67 \( 1 + (48.0 + 48.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 6.20T + 5.04e3T^{2} \)
73 \( 1 + (-37.2 + 37.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 115. iT - 6.24e3T^{2} \)
83 \( 1 + (-82.2 + 82.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + (81.9 + 81.9i)T + 9.40e3iT^{2} \)
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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

−9.229697749497615806881414840127, −8.775603648977777896608402271873, −7.66781250204810629141482091986, −6.76138438298885102844284693795, −6.45707182855921860463375549416, −4.83916789148822260937104412237, −4.32346366783776879177632365508, −2.88199581086059563100278810869, −2.06678857675431633446425301179, −0.63664434828872850653594233294, 1.25490119068667392464924193983, 2.53642602730959678550735837048, 3.68444878202062920746964069214, 4.38508936698634136366505712237, 5.47558928234450662581885923122, 6.38888273755189145671684134026, 7.37318575274827682994154009285, 8.178960846416252202346223997159, 8.935589188436498924544438171356, 9.731544913785371474216099481348

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