Properties

Label 2-200-5.4-c5-0-7
Degree $2$
Conductor $200$
Sign $-0.894 - 0.447i$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18i·3-s + 242i·7-s − 81·9-s + 656·11-s + 206i·13-s + 1.69e3i·17-s + 1.36e3·19-s − 4.35e3·21-s − 2.19e3i·23-s + 2.91e3i·27-s + 2.21e3·29-s − 1.70e3·31-s + 1.18e4i·33-s − 846i·37-s − 3.70e3·39-s + ⋯
L(s)  = 1  + 1.15i·3-s + 1.86i·7-s − 0.333·9-s + 1.63·11-s + 0.338i·13-s + 1.41i·17-s + 0.866·19-s − 2.15·21-s − 0.866i·23-s + 0.769i·27-s + 0.489·29-s − 0.317·31-s + 1.88i·33-s − 0.101i·37-s − 0.390·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.148794558\)
\(L(\frac12)\) \(\approx\) \(2.148794558\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 18iT - 243T^{2} \)
7 \( 1 - 242iT - 1.68e4T^{2} \)
11 \( 1 - 656T + 1.61e5T^{2} \)
13 \( 1 - 206iT - 3.71e5T^{2} \)
17 \( 1 - 1.69e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + 2.19e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.21e3T + 2.05e7T^{2} \)
31 \( 1 + 1.70e3T + 2.86e7T^{2} \)
37 \( 1 + 846iT - 6.93e7T^{2} \)
41 \( 1 + 1.81e3T + 1.15e8T^{2} \)
43 \( 1 + 1.05e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.20e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.25e4iT - 4.18e8T^{2} \)
59 \( 1 + 8.66e3T + 7.14e8T^{2} \)
61 \( 1 + 3.46e4T + 8.44e8T^{2} \)
67 \( 1 + 4.75e4iT - 1.35e9T^{2} \)
71 \( 1 - 948T + 1.80e9T^{2} \)
73 \( 1 - 6.31e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.65e4T + 3.07e9T^{2} \)
83 \( 1 - 8.87e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 + 3.62e4iT - 8.58e9T^{2} \)
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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

−11.96055177592254939183594249237, −11.03925985308613112581005577329, −9.813239160732917343498780638489, −9.112415629263569486379147222845, −8.453092380269791844209022712527, −6.57724988554881623945761834069, −5.63093695567149132903309164600, −4.45329224467765963188390326017, −3.35065002446976883109434462910, −1.76865738407158621027249172114, 0.74123649422388518967884185223, 1.38675957037790111408376744312, 3.36030051429201939748214060044, 4.56311255729882786575800860023, 6.31107529601396488870191056785, 7.24781720247463542924883917758, 7.58603119174489860603208742333, 9.205397388826894397048941373039, 10.17049105441251841046832650518, 11.41557916868221746021688632943

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