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Properties

Label 2-200-40.29-c3-0-19
Degree $2$
Conductor $200$
Sign $-0.717 - 0.696i$
Analytic cond. $11.8003$
Root an. cond. $3.43516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 2.59i)2-s + 6.25·3-s + (−5.44 − 5.86i)4-s + (−7.07 + 16.2i)6-s + 34.6i·7-s + (21.3 − 7.47i)8-s + 12.1·9-s − 7.91i·11-s + (−34.0 − 36.6i)12-s + 11.0·13-s + (−89.7 − 39.1i)14-s + (−4.78 + 63.8i)16-s − 57.6i·17-s + (−13.7 + 31.5i)18-s + 141. i·19-s + ⋯
L(s)  = 1  + (−0.399 + 0.916i)2-s + 1.20·3-s + (−0.680 − 0.733i)4-s + (−0.481 + 1.10i)6-s + 1.86i·7-s + (0.943 − 0.330i)8-s + 0.450·9-s − 0.216i·11-s + (−0.819 − 0.882i)12-s + 0.235·13-s + (−1.71 − 0.747i)14-s + (−0.0748 + 0.997i)16-s − 0.821i·17-s + (−0.180 + 0.412i)18-s + 1.70i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.656364 + 1.61835i\)
\(L(\frac12)\) \(\approx\) \(0.656364 + 1.61835i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 2.59i)T \)
5 \( 1 \)
good3 \( 1 - 6.25T + 27T^{2} \)
7 \( 1 - 34.6iT - 343T^{2} \)
11 \( 1 + 7.91iT - 1.33e3T^{2} \)
13 \( 1 - 11.0T + 2.19e3T^{2} \)
17 \( 1 + 57.6iT - 4.91e3T^{2} \)
19 \( 1 - 141. iT - 6.85e3T^{2} \)
23 \( 1 - 129. iT - 1.21e4T^{2} \)
29 \( 1 - 89.1iT - 2.43e4T^{2} \)
31 \( 1 + 78.3T + 2.97e4T^{2} \)
37 \( 1 - 249.T + 5.06e4T^{2} \)
41 \( 1 - 91.8T + 6.89e4T^{2} \)
43 \( 1 + 194.T + 7.95e4T^{2} \)
47 \( 1 - 72.6iT - 1.03e5T^{2} \)
53 \( 1 - 456.T + 1.48e5T^{2} \)
59 \( 1 + 341. iT - 2.05e5T^{2} \)
61 \( 1 + 217. iT - 2.26e5T^{2} \)
67 \( 1 - 529.T + 3.00e5T^{2} \)
71 \( 1 - 381.T + 3.57e5T^{2} \)
73 \( 1 + 876. iT - 3.89e5T^{2} \)
79 \( 1 - 203.T + 4.93e5T^{2} \)
83 \( 1 - 996.T + 5.71e5T^{2} \)
89 \( 1 + 172.T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−12.58119005022220929211358562746, −11.41451009595795795565967762337, −9.765509638189294297175394399629, −9.151328432547708414159526950285, −8.395536244477277269361847508428, −7.66427207388686066870119613276, −6.10664732888767218259428548814, −5.28613615351701596677110825453, −3.43183776253548376518035816870, −1.96222761613351309595017401360, 0.76705688523173370796307922037, 2.40395367579747542216131767987, 3.67955871058378274432841519558, 4.47123652838973217759234638833, 6.94016875088629650441006534166, 7.86764941911622247646899801720, 8.713042869919752592262862768048, 9.724408502003137266386021674499, 10.57503454305302761379653905446, 11.34866474984604748959606258965

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