Properties

Label 2-200-5.4-c5-0-2
Degree $2$
Conductor $200$
Sign $-0.894 + 0.447i$
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  + 20i·3-s + 24i·7-s − 157·9-s + 124·11-s + 478i·13-s + 1.19e3i·17-s − 3.04e3·19-s − 480·21-s + 184i·23-s + 1.72e3i·27-s + 3.28e3·29-s − 5.72e3·31-s + 2.48e3i·33-s − 1.03e4i·37-s − 9.56e3·39-s + ⋯
L(s)  = 1  + 1.28i·3-s + 0.185i·7-s − 0.646·9-s + 0.308·11-s + 0.784i·13-s + 1.00i·17-s − 1.93·19-s − 0.237·21-s + 0.0725i·23-s + 0.454i·27-s + 0.724·29-s − 1.07·31-s + 0.396i·33-s − 1.24i·37-s − 1.00·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8846103073\)
\(L(\frac12)\) \(\approx\) \(0.8846103073\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3 \( 1 - 20iT - 243T^{2} \)
7 \( 1 - 24iT - 1.68e4T^{2} \)
11 \( 1 - 124T + 1.61e5T^{2} \)
13 \( 1 - 478iT - 3.71e5T^{2} \)
17 \( 1 - 1.19e3iT - 1.41e6T^{2} \)
19 \( 1 + 3.04e3T + 2.47e6T^{2} \)
23 \( 1 - 184iT - 6.43e6T^{2} \)
29 \( 1 - 3.28e3T + 2.05e7T^{2} \)
31 \( 1 + 5.72e3T + 2.86e7T^{2} \)
37 \( 1 + 1.03e4iT - 6.93e7T^{2} \)
41 \( 1 + 8.88e3T + 1.15e8T^{2} \)
43 \( 1 + 9.18e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.36e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.16e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.68e4T + 7.14e8T^{2} \)
61 \( 1 + 1.84e4T + 8.44e8T^{2} \)
67 \( 1 - 1.55e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.19e4T + 1.80e9T^{2} \)
73 \( 1 + 4.88e3iT - 2.07e9T^{2} \)
79 \( 1 + 4.45e4T + 3.07e9T^{2} \)
83 \( 1 - 6.73e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.19e4T + 5.58e9T^{2} \)
97 \( 1 + 4.88e4iT - 8.58e9T^{2} \)
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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

−12.03572340098120601598174672203, −10.83012660504994142919042566429, −10.33429437644249312575827315683, −9.140979277202607640645187476456, −8.550444130005606340715493837899, −6.91489046709877375782035709337, −5.73842538736061835062800473690, −4.42902416785970411890503377463, −3.74843277682716594817605308180, −1.98498027830874329304439730333, 0.26108761409163315725902299614, 1.53897883558213299010803342532, 2.84593751831303045828783058062, 4.54273724968943560426844887269, 6.06211529109739476822646204276, 6.89938247224806911676175738282, 7.83869722148292122706082441747, 8.761513141881721603336813401091, 10.11132540666063241857693165950, 11.16993604996919892882464326307

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