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Properties

Label 2-200-5.4-c11-0-8
Degree $2$
Conductor $200$
Sign $-0.894 + 0.447i$
Analytic cond. $153.668$
Root an. cond. $12.3963$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 696. i·3-s − 7.35e4i·7-s − 3.07e5·9-s − 3.83e5·11-s + 1.15e6i·13-s + 6.51e6i·17-s + 1.39e7·19-s + 5.12e7·21-s + 1.37e6i·23-s − 9.07e7i·27-s + 7.46e7·29-s + 1.32e7·31-s − 2.66e8i·33-s + 1.67e7i·37-s − 8.06e8·39-s + ⋯
L(s)  = 1  + 1.65i·3-s − 1.65i·7-s − 1.73·9-s − 0.717·11-s + 0.865i·13-s + 1.11i·17-s + 1.29·19-s + 2.73·21-s + 0.0445i·23-s − 1.21i·27-s + 0.675·29-s + 0.0829·31-s − 1.18i·33-s + 0.0396i·37-s − 1.43·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(153.668\)
Root analytic conductor: \(12.3963\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :11/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8565789232\)
\(L(\frac12)\) \(\approx\) \(0.8565789232\)
\(L(\frac{13}{2})\) 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 \( 1 - 696. iT - 1.77e5T^{2} \)
7 \( 1 + 7.35e4iT - 1.97e9T^{2} \)
11 \( 1 + 3.83e5T + 2.85e11T^{2} \)
13 \( 1 - 1.15e6iT - 1.79e12T^{2} \)
17 \( 1 - 6.51e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.39e7T + 1.16e14T^{2} \)
23 \( 1 - 1.37e6iT - 9.52e14T^{2} \)
29 \( 1 - 7.46e7T + 1.22e16T^{2} \)
31 \( 1 - 1.32e7T + 2.54e16T^{2} \)
37 \( 1 - 1.67e7iT - 1.77e17T^{2} \)
41 \( 1 - 1.03e9T + 5.50e17T^{2} \)
43 \( 1 - 1.93e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.16e9iT - 2.47e18T^{2} \)
53 \( 1 - 4.44e8iT - 9.26e18T^{2} \)
59 \( 1 - 1.28e8T + 3.01e19T^{2} \)
61 \( 1 - 7.96e9T + 4.35e19T^{2} \)
67 \( 1 - 6.89e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.12e10T + 2.31e20T^{2} \)
73 \( 1 - 3.34e9iT - 3.13e20T^{2} \)
79 \( 1 + 5.36e10T + 7.47e20T^{2} \)
83 \( 1 + 6.31e10iT - 1.28e21T^{2} \)
89 \( 1 + 9.62e10T + 2.77e21T^{2} \)
97 \( 1 - 4.37e10iT - 7.15e21T^{2} \)
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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

−10.72075995258545223584993685455, −10.14926358557688431290826411106, −9.396511663214024326015265417465, −8.174678177463881323392332432430, −7.09593921504717769853294609309, −5.69121945706175746017990868829, −4.50949493372011150233607435633, −4.01597623716675698287438478954, −2.98027248315862085449081067739, −1.17584941275648228869495108951, 0.18045297029743166718863332116, 1.18838479210467286673653522501, 2.48392457450707005390787738423, 2.86412336350785449634464107283, 5.26073878123809510787370663016, 5.78044247621180367837471060008, 6.98334359644548105306869377393, 7.85821688012521427417903440598, 8.618360722088657985562411907605, 9.728093769929389519040964745972

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