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Properties

Label 2-200-8.5-c5-0-16
Degree $2$
Conductor $200$
Sign $0.973 - 0.230i$
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  + (−5.63 + 0.438i)2-s − 17.0i·3-s + (31.6 − 4.94i)4-s + (7.47 + 96.1i)6-s − 191.·7-s + (−176. + 41.7i)8-s − 47.4·9-s − 669. i·11-s + (−84.3 − 538. i)12-s + 1.15e3i·13-s + (1.07e3 − 83.8i)14-s + (975. − 312. i)16-s − 1.03e3·17-s + (267. − 20.8i)18-s + 1.51e3i·19-s + ⋯
L(s)  = 1  + (−0.996 + 0.0775i)2-s − 1.09i·3-s + (0.987 − 0.154i)4-s + (0.0847 + 1.08i)6-s − 1.47·7-s + (−0.973 + 0.230i)8-s − 0.195·9-s − 1.66i·11-s + (−0.169 − 1.08i)12-s + 1.89i·13-s + (1.47 − 0.114i)14-s + (0.952 − 0.305i)16-s − 0.872·17-s + (0.194 − 0.0151i)18-s + 0.964i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.973 - 0.230i$
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: \((2,\ 200,\ (\ :5/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6795184326\)
\(L(\frac12)\) \(\approx\) \(0.6795184326\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.63 - 0.438i)T \)
5 \( 1 \)
good3 \( 1 + 17.0iT - 243T^{2} \)
7 \( 1 + 191.T + 1.68e4T^{2} \)
11 \( 1 + 669. iT - 1.61e5T^{2} \)
13 \( 1 - 1.15e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 1.51e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.96e3T + 6.43e6T^{2} \)
29 \( 1 - 2.02e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.58e3T + 2.86e7T^{2} \)
37 \( 1 + 1.10e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.25e3T + 1.15e8T^{2} \)
43 \( 1 - 3.00e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.62e4T + 2.29e8T^{2} \)
53 \( 1 - 2.09e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.29e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.30e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.49e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.15e4T + 1.80e9T^{2} \)
73 \( 1 - 6.46e4T + 2.07e9T^{2} \)
79 \( 1 - 1.65e4T + 3.07e9T^{2} \)
83 \( 1 + 4.83e4iT - 3.93e9T^{2} \)
89 \( 1 - 7.77e4T + 5.58e9T^{2} \)
97 \( 1 - 1.32e5T + 8.58e9T^{2} \)
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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

−11.57129502951312683850998896165, −10.63219274563416085391471481975, −9.306524526829795215493229923737, −8.781487872456312832573345794317, −7.42947872672419337819411542379, −6.58345029541796863371444477415, −6.06955304651116006133502857572, −3.54049653747015514201432235602, −2.15340530735037034030322964280, −0.841439559803710343574782909826, 0.38833130736843324173588894809, 2.51989668208505619897227659257, 3.64441595853234077448889179788, 5.16094511727856981330258892312, 6.63130742369550878149703069895, 7.49144470509984157498396143784, 9.003883026245802626014264500138, 9.664419339232727994854530457360, 10.26344127390696450421904133614, 11.03271347021985354826827736608

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