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Properties

Label 2-200-40.3-c3-0-16
Degree $2$
Conductor $200$
Sign $-0.768 - 0.640i$
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  + (−1.14 + 2.58i)2-s + (4.39 + 4.39i)3-s + (−5.38 − 5.91i)4-s + (−16.3 + 6.34i)6-s + (18.5 + 18.5i)7-s + (21.4 − 7.16i)8-s + 11.6i·9-s + 59.8·11-s + (2.33 − 49.6i)12-s + (−37.6 + 37.6i)13-s + (−69.3 + 26.8i)14-s + (−6.00 + 63.7i)16-s + (−7.72 + 7.72i)17-s + (−30.1 − 13.3i)18-s + 10.5i·19-s + ⋯
L(s)  = 1  + (−0.404 + 0.914i)2-s + (0.845 + 0.845i)3-s + (−0.673 − 0.739i)4-s + (−1.11 + 0.431i)6-s + (1.00 + 1.00i)7-s + (0.948 − 0.316i)8-s + 0.431i·9-s + 1.64·11-s + (0.0561 − 1.19i)12-s + (−0.804 + 0.804i)13-s + (−1.32 + 0.512i)14-s + (−0.0938 + 0.995i)16-s + (−0.110 + 0.110i)17-s + (−0.394 − 0.174i)18-s + 0.127i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.768 - 0.640i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ -0.768 - 0.640i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.672751 + 1.85739i\)
\(L(\frac12)\) \(\approx\) \(0.672751 + 1.85739i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 2.58i)T \)
5 \( 1 \)
good3 \( 1 + (-4.39 - 4.39i)T + 27iT^{2} \)
7 \( 1 + (-18.5 - 18.5i)T + 343iT^{2} \)
11 \( 1 - 59.8T + 1.33e3T^{2} \)
13 \( 1 + (37.6 - 37.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (7.72 - 7.72i)T - 4.91e3iT^{2} \)
19 \( 1 - 10.5iT - 6.85e3T^{2} \)
23 \( 1 + (32.9 - 32.9i)T - 1.21e4iT^{2} \)
29 \( 1 + 90.1T + 2.43e4T^{2} \)
31 \( 1 - 35.5iT - 2.97e4T^{2} \)
37 \( 1 + (172. + 172. i)T + 5.06e4iT^{2} \)
41 \( 1 - 70.6T + 6.89e4T^{2} \)
43 \( 1 + (62.6 + 62.6i)T + 7.95e4iT^{2} \)
47 \( 1 + (-102. - 102. i)T + 1.03e5iT^{2} \)
53 \( 1 + (227. - 227. i)T - 1.48e5iT^{2} \)
59 \( 1 - 466. iT - 2.05e5T^{2} \)
61 \( 1 + 527. iT - 2.26e5T^{2} \)
67 \( 1 + (-175. + 175. i)T - 3.00e5iT^{2} \)
71 \( 1 - 185. iT - 3.57e5T^{2} \)
73 \( 1 + (-206. - 206. i)T + 3.89e5iT^{2} \)
79 \( 1 - 464.T + 4.93e5T^{2} \)
83 \( 1 + (194. + 194. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.46e3iT - 7.04e5T^{2} \)
97 \( 1 + (-1.03e3 + 1.03e3i)T - 9.12e5iT^{2} \)
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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

−12.29418692365155931689602658299, −11.32266391996892497971437337067, −9.911715224623540742537035967948, −9.052904728978172565373436310590, −8.749951353282481597365266329757, −7.47635399569270149319871368220, −6.20528748144681314958405003447, −4.89166137965108186043987173639, −3.92443482369748893211141930727, −1.84319221356256395392429023035, 0.995595093630575829600226295027, 2.04798617389197501860591666731, 3.54597939846744159776042523080, 4.74804274646331783708837345095, 6.97208011145573141176670575886, 7.77903345778699736250113384022, 8.550729100035835442011483763537, 9.625925032488073734571155453963, 10.71454641337943415070380408838, 11.65373382648561989518750213865

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