Properties

Label 2-200-5.3-c4-0-16
Degree $2$
Conductor $200$
Sign $-0.229 + 0.973i$
Analytic cond. $20.6739$
Root an. cond. $4.54686$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 3i)3-s + (42.8 − 42.8i)7-s − 63i·9-s − 181.·11-s + (−25.1 − 25.1i)13-s + (−160. + 160. i)17-s − 80.2i·19-s + 257.·21-s + (−324. − 324. i)23-s + (432 − 432i)27-s − 1.37e3i·29-s − 151.·31-s + (−545. − 545. i)33-s + (1.42e3 − 1.42e3i)37-s − 150. i·39-s + ⋯
L(s)  = 1  + (0.333 + 0.333i)3-s + (0.874 − 0.874i)7-s − 0.777i·9-s − 1.50·11-s + (−0.148 − 0.148i)13-s + (−0.555 + 0.555i)17-s − 0.222i·19-s + 0.583·21-s + (−0.613 − 0.613i)23-s + (0.592 − 0.592i)27-s − 1.63i·29-s − 0.157·31-s + (−0.500 − 0.500i)33-s + (1.04 − 1.04i)37-s − 0.0992i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(20.6739\)
Root analytic conductor: \(4.54686\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :2),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.438218833\)
\(L(\frac12)\) \(\approx\) \(1.438218833\)
\(L(3)\) 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 \( 1 + (-3 - 3i)T + 81iT^{2} \)
7 \( 1 + (-42.8 + 42.8i)T - 2.40e3iT^{2} \)
11 \( 1 + 181.T + 1.46e4T^{2} \)
13 \( 1 + (25.1 + 25.1i)T + 2.85e4iT^{2} \)
17 \( 1 + (160. - 160. i)T - 8.35e4iT^{2} \)
19 \( 1 + 80.2iT - 1.30e5T^{2} \)
23 \( 1 + (324. + 324. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.37e3iT - 7.07e5T^{2} \)
31 \( 1 + 151.T + 9.23e5T^{2} \)
37 \( 1 + (-1.42e3 + 1.42e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 3.01e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.61e3 - 1.61e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-397. + 397. i)T - 4.87e6iT^{2} \)
53 \( 1 + (923. + 923. i)T + 7.89e6iT^{2} \)
59 \( 1 + 4.54e3iT - 1.21e7T^{2} \)
61 \( 1 - 631.T + 1.38e7T^{2} \)
67 \( 1 + (-2.68e3 + 2.68e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 4.54e3T + 2.54e7T^{2} \)
73 \( 1 + (5.85e3 + 5.85e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 9.42e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.39e3 - 8.39e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 8.36e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.01e3 - 1.01e3i)T - 8.85e7iT^{2} \)
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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

−11.31933351950703979539687208660, −10.51004168328682600026441304060, −9.659598357452507196327398703784, −8.322277607744480621689023344041, −7.68013168951677639421964802223, −6.29369626084751354354739535337, −4.87126972656200099504434797386, −3.88065279264343908263429237676, −2.33920060926883574627003423580, −0.46187094050127873801729262286, 1.82681275201194259259019278560, 2.80370490822258761139364600802, 4.82249340129072188896161908269, 5.53475547333505282182806963261, 7.22267038856715602964640693302, 8.075542966475353554406489967132, 8.819643624429639821938974418892, 10.21376858691975925274165323556, 11.11744571125405199605352324868, 12.07687947622378540720028406814

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