Properties

Label 2-200-5.4-c3-0-13
Degree $2$
Conductor $200$
Sign $-0.447 - 0.894i$
Analytic cond. $11.8003$
Root an. cond. $3.43516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9i·3-s + 26i·7-s − 54·9-s − 59·11-s − 28i·13-s + 5i·17-s − 109·19-s + 234·21-s + 194i·23-s + 243i·27-s + 32·29-s + 10·31-s + 531i·33-s − 198i·37-s − 252·39-s + ⋯
L(s)  = 1  − 1.73i·3-s + 1.40i·7-s − 2·9-s − 1.61·11-s − 0.597i·13-s + 0.0713i·17-s − 1.31·19-s + 2.43·21-s + 1.75i·23-s + 1.73i·27-s + 0.204·29-s + 0.0579·31-s + 2.80i·33-s − 0.879i·37-s − 1.03·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 9iT - 27T^{2} \)
7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 + 59T + 1.33e3T^{2} \)
13 \( 1 + 28iT - 2.19e3T^{2} \)
17 \( 1 - 5iT - 4.91e3T^{2} \)
19 \( 1 + 109T + 6.85e3T^{2} \)
23 \( 1 - 194iT - 1.21e4T^{2} \)
29 \( 1 - 32T + 2.43e4T^{2} \)
31 \( 1 - 10T + 2.97e4T^{2} \)
37 \( 1 + 198iT - 5.06e4T^{2} \)
41 \( 1 - 117T + 6.89e4T^{2} \)
43 \( 1 + 388iT - 7.95e4T^{2} \)
47 \( 1 + 68iT - 1.03e5T^{2} \)
53 \( 1 - 18iT - 1.48e5T^{2} \)
59 \( 1 + 392T + 2.05e5T^{2} \)
61 \( 1 + 710T + 2.26e5T^{2} \)
67 \( 1 + 253iT - 3.00e5T^{2} \)
71 \( 1 + 612T + 3.57e5T^{2} \)
73 \( 1 - 549iT - 3.89e5T^{2} \)
79 \( 1 + 414T + 4.93e5T^{2} \)
83 \( 1 - 121iT - 5.71e5T^{2} \)
89 \( 1 - 81T + 7.04e5T^{2} \)
97 \( 1 + 1.50e3iT - 9.12e5T^{2} \)
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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

−11.67348721665217986390393719824, −10.57688960264807055982568372232, −8.979985282582842188120706929513, −8.104698253017225594076831966140, −7.38160533140963653868149082931, −6.01862108603600452737511220259, −5.40348380460050031519800036264, −2.83505152557841791525048598942, −1.93969814527912622473200520647, 0, 2.86651536518036680132163365638, 4.30213702937028984728472696706, 4.74456189187046857897397817155, 6.31954877183554949997475755517, 7.82334988108034490830774446384, 8.845914893732051975166137237585, 10.11515541157881633164754281975, 10.49402007663718025376895868752, 11.15393938352522419346794942557

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