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Properties

Label 2-200-40.3-c3-0-0
Degree $2$
Conductor $200$
Sign $-0.0621 - 0.998i$
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  + (−0.995 − 2.64i)2-s + (0.102 + 0.102i)3-s + (−6.01 + 5.27i)4-s + (0.169 − 0.373i)6-s + (−15.5 − 15.5i)7-s + (19.9 + 10.6i)8-s − 26.9i·9-s + 20.0·11-s + (−1.15 − 0.0762i)12-s + (−9.16 + 9.16i)13-s + (−25.6 + 56.6i)14-s + (8.39 − 63.4i)16-s + (−74.0 + 74.0i)17-s + (−71.4 + 26.8i)18-s + 111. i·19-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)2-s + (0.0197 + 0.0197i)3-s + (−0.752 + 0.659i)4-s + (0.0115 − 0.0254i)6-s + (−0.839 − 0.839i)7-s + (0.881 + 0.471i)8-s − 0.999i·9-s + 0.550·11-s + (−0.0278 − 0.00183i)12-s + (−0.195 + 0.195i)13-s + (−0.490 + 1.08i)14-s + (0.131 − 0.991i)16-s + (−1.05 + 1.05i)17-s + (−0.935 + 0.351i)18-s + 1.35i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0621 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0621 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.0621 - 0.998i$
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: \((2,\ 200,\ (\ :3/2),\ -0.0621 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0997785 + 0.106186i\)
\(L(\frac12)\) \(\approx\) \(0.0997785 + 0.106186i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 + 2.64i)T \)
5 \( 1 \)
good3 \( 1 + (-0.102 - 0.102i)T + 27iT^{2} \)
7 \( 1 + (15.5 + 15.5i)T + 343iT^{2} \)
11 \( 1 - 20.0T + 1.33e3T^{2} \)
13 \( 1 + (9.16 - 9.16i)T - 2.19e3iT^{2} \)
17 \( 1 + (74.0 - 74.0i)T - 4.91e3iT^{2} \)
19 \( 1 - 111. iT - 6.85e3T^{2} \)
23 \( 1 + (75.1 - 75.1i)T - 1.21e4iT^{2} \)
29 \( 1 + 203.T + 2.43e4T^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-64.2 - 64.2i)T + 5.06e4iT^{2} \)
41 \( 1 + 41.9T + 6.89e4T^{2} \)
43 \( 1 + (133. + 133. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-280. - 280. i)T + 1.03e5iT^{2} \)
53 \( 1 + (14.0 - 14.0i)T - 1.48e5iT^{2} \)
59 \( 1 + 774. iT - 2.05e5T^{2} \)
61 \( 1 + 159. iT - 2.26e5T^{2} \)
67 \( 1 + (425. - 425. i)T - 3.00e5iT^{2} \)
71 \( 1 - 843. iT - 3.57e5T^{2} \)
73 \( 1 + (721. + 721. i)T + 3.89e5iT^{2} \)
79 \( 1 + 249.T + 4.93e5T^{2} \)
83 \( 1 + (-294. - 294. i)T + 5.71e5iT^{2} \)
89 \( 1 + 316. iT - 7.04e5T^{2} \)
97 \( 1 + (-142. + 142. i)T - 9.12e5iT^{2} \)
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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

−12.25374413563097286042377308553, −11.25713676914884114743703140874, −10.19451900015418764106585062876, −9.544115678751792617271446108156, −8.560686924577439561124528198632, −7.25111423207328467824976019140, −6.09997980406538892967764753179, −4.09536344588858965367374972272, −3.50866836749946520363920059593, −1.58472738069017272034950002893, 0.07041441499340488992569516442, 2.42866720100338050449074281906, 4.45349369697873239181627561152, 5.57162579163709424318062037768, 6.63843752355450465212537566431, 7.57203392879233005052876349209, 8.876109858762901698898640248285, 9.376058474562056110909154925543, 10.57993193444807916655981946691, 11.70929042221307176915637575515

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