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Properties

Label 2-200-8.5-c5-0-90
Degree $2$
Conductor $200$
Sign $0.821 - 0.570i$
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  + (1.78 − 5.36i)2-s − 29.6i·3-s + (−25.6 − 19.1i)4-s + (−158. − 52.8i)6-s + 90.3·7-s + (−148. + 103. i)8-s − 633.·9-s − 181. i·11-s + (−567. + 758. i)12-s + 455. i·13-s + (161. − 484. i)14-s + (289. + 982. i)16-s + 615.·17-s + (−1.13e3 + 3.39e3i)18-s + 2.49e3i·19-s + ⋯
L(s)  = 1  + (0.315 − 0.948i)2-s − 1.89i·3-s + (−0.800 − 0.598i)4-s + (−1.80 − 0.599i)6-s + 0.696·7-s + (−0.821 + 0.570i)8-s − 2.60·9-s − 0.453i·11-s + (−1.13 + 1.52i)12-s + 0.747i·13-s + (0.219 − 0.661i)14-s + (0.282 + 0.959i)16-s + 0.516·17-s + (−0.822 + 2.47i)18-s + 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.821 - 0.570i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ 0.821 - 0.570i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4737553576\)
\(L(\frac12)\) \(\approx\) \(0.4737553576\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78 + 5.36i)T \)
5 \( 1 \)
good3 \( 1 + 29.6iT - 243T^{2} \)
7 \( 1 - 90.3T + 1.68e4T^{2} \)
11 \( 1 + 181. iT - 1.61e5T^{2} \)
13 \( 1 - 455. iT - 3.71e5T^{2} \)
17 \( 1 - 615.T + 1.41e6T^{2} \)
19 \( 1 - 2.49e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.42e3T + 6.43e6T^{2} \)
29 \( 1 + 7.65e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.76e3T + 2.86e7T^{2} \)
37 \( 1 - 4.80e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.88e3T + 1.15e8T^{2} \)
43 \( 1 + 8.26e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.72e3T + 2.29e8T^{2} \)
53 \( 1 + 3.23e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.08e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.32e4iT - 8.44e8T^{2} \)
67 \( 1 - 9.64e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.30e4T + 1.80e9T^{2} \)
73 \( 1 + 1.85e4T + 2.07e9T^{2} \)
79 \( 1 - 4.42e4T + 3.07e9T^{2} \)
83 \( 1 - 8.00e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.16e4T + 5.58e9T^{2} \)
97 \( 1 - 2.22e4T + 8.58e9T^{2} \)
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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

−11.23339117053577808702626433278, −9.827439828004990414183051921703, −8.370979219174774091163080607853, −7.86204135901142712257340102165, −6.32803161519409592568524409163, −5.53728256399413746293677837977, −3.73870076719015714356116893303, −2.14009628735572284147483914344, −1.50705365003963884146911780226, −0.12744145511716998631373247800, 3.07298722385359718139960286508, 4.22464110494665354232187327600, 5.01619616709867535592445422720, 5.81830486778996244664871343986, 7.51129633031039289678654280319, 8.610165521063929152187401412167, 9.369963533473970808182310396353, 10.35040746485606516591445485099, 11.28182721227493934640410950959, 12.47684876182014469245366780778

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