Properties

Label 2-1200-15.14-c2-0-20
Degree $2$
Conductor $1200$
Sign $-0.541 - 0.840i$
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  + (−1.52 + 2.58i)3-s + 7.48i·7-s + (−4.32 − 7.89i)9-s − 8.48i·11-s + 10i·13-s + 30.3·17-s + 26.9·19-s + (−19.3 − 11.4i)21-s − 9.17·23-s + (26.9 + 0.905i)27-s + 26.8i·29-s − 8·31-s + (21.9 + 12.9i)33-s + 15.9i·37-s + (−25.8 − 15.2i)39-s + ⋯
L(s)  = 1  + (−0.509 + 0.860i)3-s + 1.06i·7-s + (−0.480 − 0.876i)9-s − 0.771i·11-s + 0.769i·13-s + 1.78·17-s + 1.41·19-s + (−0.920 − 0.545i)21-s − 0.398·23-s + (0.999 + 0.0335i)27-s + 0.925i·29-s − 0.258·31-s + (0.663 + 0.393i)33-s + 0.431i·37-s + (−0.661 − 0.392i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.541 - 0.840i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.482531906\)
\(L(\frac12)\) \(\approx\) \(1.482531906\)
\(L(2)\) 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 \( 1 + (1.52 - 2.58i)T \)
5 \( 1 \)
good7 \( 1 - 7.48iT - 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 - 30.3T + 289T^{2} \)
19 \( 1 - 26.9T + 361T^{2} \)
23 \( 1 + 9.17T + 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 + 8T + 961T^{2} \)
37 \( 1 - 15.9iT - 1.36e3T^{2} \)
41 \( 1 - 47.3iT - 1.68e3T^{2} \)
43 \( 1 + 14.4iT - 1.84e3T^{2} \)
47 \( 1 - 45.8T + 2.20e3T^{2} \)
53 \( 1 + 30.3T + 2.80e3T^{2} \)
59 \( 1 + 24.0iT - 3.48e3T^{2} \)
61 \( 1 + 53.9T + 3.72e3T^{2} \)
67 \( 1 - 110. iT - 4.48e3T^{2} \)
71 \( 1 + 15.5iT - 5.04e3T^{2} \)
73 \( 1 + 87.9iT - 5.32e3T^{2} \)
79 \( 1 + 46.9T + 6.24e3T^{2} \)
83 \( 1 + 26.1T + 6.88e3T^{2} \)
89 \( 1 + 60.7iT - 7.92e3T^{2} \)
97 \( 1 - 36.0iT - 9.40e3T^{2} \)
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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

−9.748939431938225921699471690748, −9.185848908517888182517794370168, −8.413135451234823884473083240498, −7.38275845567492122287790134144, −6.14901259767645982076694320352, −5.59822083449066670367393085132, −4.89994570456839981131003422651, −3.61448952250376333798368329562, −2.92011290425297270395796609749, −1.19092333433919579314979078978, 0.55906678386175963430665948398, 1.47856466779378053281949219108, 2.90961931020816535615771411109, 4.02912101898299775868238018810, 5.25358182190563125609191375742, 5.82976337771118473492717497437, 7.02112233109406102646678250942, 7.59923309875272852771835526908, 7.993799972017219232069614218412, 9.467661786170570569853435763037

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