Properties

Label 2-200-5.3-c4-0-12
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 + (−64.8 + 64.8i)7-s − 63i·9-s + 33.7·11-s + (−132. − 132. i)13-s + (270. − 270. i)17-s − 295. i·19-s − 389.·21-s + (−1.44 − 1.44i)23-s + (432 − 432i)27-s − 297. i·29-s − 1.22e3·31-s + (101. + 101. i)33-s + (−402. + 402. i)37-s − 797. i·39-s + ⋯
L(s)  = 1  + (0.333 + 0.333i)3-s + (−1.32 + 1.32i)7-s − 0.777i·9-s + 0.278·11-s + (−0.786 − 0.786i)13-s + (0.935 − 0.935i)17-s − 0.819i·19-s − 0.882·21-s + (−0.00273 − 0.00273i)23-s + (0.592 − 0.592i)27-s − 0.353i·29-s − 1.27·31-s + (0.0928 + 0.0928i)33-s + (−0.293 + 0.293i)37-s − 0.524i·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

\(L(\frac{5}{2})\) \(\approx\) \(0.8391711304\)
\(L(\frac12)\) \(\approx\) \(0.8391711304\)
\(L(3)\) 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 \( 1 + (64.8 - 64.8i)T - 2.40e3iT^{2} \)
11 \( 1 - 33.7T + 1.46e4T^{2} \)
13 \( 1 + (132. + 132. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-270. + 270. i)T - 8.35e4iT^{2} \)
19 \( 1 + 295. iT - 1.30e5T^{2} \)
23 \( 1 + (1.44 + 1.44i)T + 2.79e5iT^{2} \)
29 \( 1 + 297. iT - 7.07e5T^{2} \)
31 \( 1 + 1.22e3T + 9.23e5T^{2} \)
37 \( 1 + (402. - 402. i)T - 1.87e6iT^{2} \)
41 \( 1 - 860.T + 2.82e6T^{2} \)
43 \( 1 + (753. + 753. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-2.22e3 + 2.22e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.53e3 + 2.53e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.42e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.69e3T + 1.38e7T^{2} \)
67 \( 1 + (-318. + 318. i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.42e3T + 2.54e7T^{2} \)
73 \( 1 + (3.05e3 + 3.05e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 2.63e3iT - 3.89e7T^{2} \)
83 \( 1 + (225. + 225. i)T + 4.74e7iT^{2} \)
89 \( 1 - 4.98e3iT - 6.27e7T^{2} \)
97 \( 1 + (4.89e3 - 4.89e3i)T - 8.85e7iT^{2} \)
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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

−11.84167740652236874579533003091, −10.21254285992670388123008799763, −9.394276384009051729124803040364, −8.902098910257098687576650753092, −7.36112016982547799977491814140, −6.21169657175480113294324945555, −5.20919109537640567259281003386, −3.44735659788003820365513379003, −2.65640934851210244953993566660, −0.27973126508083906074520844002, 1.53516113652815665322984057231, 3.22289677007674935477358898514, 4.31388281171623696253127698246, 5.98144964697240152300765097885, 7.13931694838132321666124239493, 7.76027080348157763357212348764, 9.221513951678434287420271754064, 10.12602489292295835973269073975, 10.87335582079514409019861022036, 12.41081273464281095165987009228

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