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Properties

Label 2-200-25.16-c1-0-3
Degree $2$
Conductor $200$
Sign $0.988 - 0.152i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.372 + 1.14i)3-s + (0.388 − 2.20i)5-s + 1.59·7-s + (1.25 − 0.909i)9-s + (2.86 + 2.07i)11-s + (−2.38 + 1.73i)13-s + (2.66 − 0.374i)15-s + (−0.357 + 1.10i)17-s + (−0.866 + 2.66i)19-s + (0.595 + 1.83i)21-s + (−2.55 − 1.85i)23-s + (−4.69 − 1.71i)25-s + (4.43 + 3.22i)27-s + (−3.04 − 9.35i)29-s + (0.572 − 1.76i)31-s + ⋯
L(s)  = 1  + (0.215 + 0.661i)3-s + (0.173 − 0.984i)5-s + 0.604·7-s + (0.417 − 0.303i)9-s + (0.862 + 0.626i)11-s + (−0.661 + 0.480i)13-s + (0.689 − 0.0967i)15-s + (−0.0867 + 0.267i)17-s + (−0.198 + 0.611i)19-s + (0.129 + 0.400i)21-s + (−0.533 − 0.387i)23-s + (−0.939 − 0.342i)25-s + (0.853 + 0.619i)27-s + (−0.564 − 1.73i)29-s + (0.102 − 0.316i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39349 + 0.107043i\)
\(L(\frac12)\) \(\approx\) \(1.39349 + 0.107043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.388 + 2.20i)T \)
good3 \( 1 + (-0.372 - 1.14i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
11 \( 1 + (-2.86 - 2.07i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.38 - 1.73i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.357 - 1.10i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.866 - 2.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.55 + 1.85i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (3.04 + 9.35i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.572 + 1.76i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.63 - 2.64i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.49 + 4.71i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-0.728 - 2.24i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.15 - 6.62i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.1 - 8.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.96 - 1.42i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.27 - 13.1i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.07 + 3.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.63 - 1.91i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.06 + 9.43i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.97 + 12.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-8.85 - 6.43i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.21 + 6.81i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.29756576266228559096599897791, −11.75954368865430374268808874057, −10.23247405729007120387497423990, −9.535694718585400790298095404276, −8.707855664000216086481659646774, −7.55016655328727084398392162000, −6.11945719690795383140231338140, −4.65037777765023817913635162256, −4.08300441396124652641572898281, −1.77241804165931223893860070844, 1.84067313449407131515028366664, 3.32104834500245463833076492960, 5.01905534605606843176950914243, 6.49229522314692335529504909700, 7.27116648648185684774591349001, 8.219085130511417675080259703459, 9.502353679756964648327547674015, 10.63976945841934551205632404732, 11.40464189044520341917795873479, 12.46575933286234271929810553239

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