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Properties

Label 2-200-40.29-c5-0-27
Degree $2$
Conductor $200$
Sign $-0.837 - 0.545i$
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  + (−0.214 + 5.65i)2-s + 18.7·3-s + (−31.9 − 2.42i)4-s + (−4.03 + 106. i)6-s − 107. i·7-s + (20.5 − 179. i)8-s + 109.·9-s + 272. i·11-s + (−599. − 45.5i)12-s − 198.·13-s + (607. + 23.0i)14-s + (1.01e3 + 154. i)16-s + 2.06e3i·17-s + (−23.5 + 621. i)18-s + 1.89e3i·19-s + ⋯
L(s)  = 1  + (−0.0379 + 0.999i)2-s + 1.20·3-s + (−0.997 − 0.0757i)4-s + (−0.0457 + 1.20i)6-s − 0.829i·7-s + (0.113 − 0.993i)8-s + 0.452·9-s + 0.678i·11-s + (−1.20 − 0.0913i)12-s − 0.325·13-s + (0.828 + 0.0314i)14-s + (0.988 + 0.151i)16-s + 1.73i·17-s + (−0.0171 + 0.452i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.837 - 0.545i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -0.837 - 0.545i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.000178983\)
\(L(\frac12)\) \(\approx\) \(2.000178983\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.214 - 5.65i)T \)
5 \( 1 \)
good3 \( 1 - 18.7T + 243T^{2} \)
7 \( 1 + 107. iT - 1.68e4T^{2} \)
11 \( 1 - 272. iT - 1.61e5T^{2} \)
13 \( 1 + 198.T + 3.71e5T^{2} \)
17 \( 1 - 2.06e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.89e3iT - 2.47e6T^{2} \)
23 \( 1 - 987. iT - 6.43e6T^{2} \)
29 \( 1 - 8.01e3iT - 2.05e7T^{2} \)
31 \( 1 - 827.T + 2.86e7T^{2} \)
37 \( 1 - 9.42e3T + 6.93e7T^{2} \)
41 \( 1 + 8.22e3T + 1.15e8T^{2} \)
43 \( 1 - 9.30e3T + 1.47e8T^{2} \)
47 \( 1 - 1.38e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.77e4T + 4.18e8T^{2} \)
59 \( 1 + 2.51e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.64e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.85e4T + 1.35e9T^{2} \)
71 \( 1 + 7.10e4T + 1.80e9T^{2} \)
73 \( 1 + 1.86e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.55e4T + 3.07e9T^{2} \)
83 \( 1 + 1.25e5T + 3.93e9T^{2} \)
89 \( 1 + 3.03e4T + 5.58e9T^{2} \)
97 \( 1 - 1.56e4iT - 8.58e9T^{2} \)
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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

−12.45846450131474936915348055348, −10.55887415077192994466410158691, −9.757278627514257709056053828517, −8.718496984805164292501145731676, −7.910781529723529163151766850614, −7.17015136861881021997026954856, −5.87544460326345948155424841355, −4.34696826202529040576947244123, −3.46121859857541637989201904099, −1.55134694009352904798007621802, 0.53406277465308975867335877180, 2.48318532069844863634194154227, 2.81187431058671120894161946817, 4.32279561047950154368772725521, 5.61685796608592472252090006049, 7.47697667261892096017800574743, 8.626350946465473828877933587608, 9.112161537874882622053406471496, 9.991155544913944979439651214591, 11.41741951323517209949600270814

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