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Properties

Label 2-200-40.29-c5-0-59
Degree $2$
Conductor $200$
Sign $0.403 - 0.915i$
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  + (4.94 + 2.74i)2-s + 23.5·3-s + (16.8 + 27.1i)4-s + (116. + 64.7i)6-s + 39.5i·7-s + (8.81 + 180. i)8-s + 311.·9-s − 236. i·11-s + (397. + 639. i)12-s + 942.·13-s + (−108. + 195. i)14-s + (−453. + 918. i)16-s − 1.10e3i·17-s + (1.53e3 + 856. i)18-s + 2.31e3i·19-s + ⋯
L(s)  = 1  + (0.874 + 0.485i)2-s + 1.51·3-s + (0.527 + 0.849i)4-s + (1.32 + 0.733i)6-s + 0.305i·7-s + (0.0486 + 0.998i)8-s + 1.28·9-s − 0.589i·11-s + (0.797 + 1.28i)12-s + 1.54·13-s + (−0.148 + 0.266i)14-s + (−0.442 + 0.896i)16-s − 0.925i·17-s + (1.12 + 0.622i)18-s + 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.403 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.403 - 0.915i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ 0.403 - 0.915i)\)

Particular Values

\(L(3)\) \(\approx\) \(5.974087753\)
\(L(\frac12)\) \(\approx\) \(5.974087753\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.94 - 2.74i)T \)
5 \( 1 \)
good3 \( 1 - 23.5T + 243T^{2} \)
7 \( 1 - 39.5iT - 1.68e4T^{2} \)
11 \( 1 + 236. iT - 1.61e5T^{2} \)
13 \( 1 - 942.T + 3.71e5T^{2} \)
17 \( 1 + 1.10e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.31e3iT - 2.47e6T^{2} \)
23 \( 1 - 861. iT - 6.43e6T^{2} \)
29 \( 1 - 1.66e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.18e3T + 2.86e7T^{2} \)
37 \( 1 - 1.45e4T + 6.93e7T^{2} \)
41 \( 1 + 2.00e4T + 1.15e8T^{2} \)
43 \( 1 + 5.24e3T + 1.47e8T^{2} \)
47 \( 1 + 2.16e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.49e4T + 4.18e8T^{2} \)
59 \( 1 + 4.24e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.11e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.78e4T + 1.35e9T^{2} \)
71 \( 1 - 4.23e4T + 1.80e9T^{2} \)
73 \( 1 + 1.17e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.98e4T + 3.07e9T^{2} \)
83 \( 1 + 6.45e3T + 3.93e9T^{2} \)
89 \( 1 + 2.56e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5iT - 8.58e9T^{2} \)
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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

−12.01111849326713233366252608296, −10.95287543930087956953689542476, −9.433711677132999338213710325691, −8.426986814726835929080934231098, −7.919343978201733881988276299687, −6.58756314737593498305679988668, −5.43196443524764680544740084551, −3.79536470089855203791602441131, −3.21636302404407496970634896658, −1.82940664296443008607828149098, 1.32122389604748337673496048353, 2.52550865256725532870674391186, 3.60433716391253172123757477085, 4.47041793696328614695262569853, 6.14239880411160770018495813406, 7.29936796933131473106335123276, 8.510549476037181597073523464986, 9.412765549620005320602485278627, 10.49654078174791012693643698352, 11.39991002522366641352963003020

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