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Properties

Label 2-200-40.29-c5-0-40
Degree $2$
Conductor $200$
Sign $0.986 - 0.162i$
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  + (1.74 − 5.37i)2-s + 25.2·3-s + (−25.8 − 18.8i)4-s + (44.2 − 135. i)6-s + 185. i·7-s + (−146. + 106. i)8-s + 395.·9-s + 574. i·11-s + (−653. − 475. i)12-s + 65.8·13-s + (996. + 324. i)14-s + (315. + 974. i)16-s + 1.96e3i·17-s + (691. − 2.12e3i)18-s + 611. i·19-s + ⋯
L(s)  = 1  + (0.309 − 0.950i)2-s + 1.62·3-s + (−0.808 − 0.588i)4-s + (0.501 − 1.54i)6-s + 1.42i·7-s + (−0.809 + 0.587i)8-s + 1.62·9-s + 1.43i·11-s + (−1.31 − 0.953i)12-s + 0.108·13-s + (1.35 + 0.441i)14-s + (0.307 + 0.951i)16-s + 1.65i·17-s + (0.503 − 1.54i)18-s + 0.388i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.986 - 0.162i$
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: \((2,\ 200,\ (\ :5/2),\ 0.986 - 0.162i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.497804899\)
\(L(\frac12)\) \(\approx\) \(3.497804899\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.74 + 5.37i)T \)
5 \( 1 \)
good3 \( 1 - 25.2T + 243T^{2} \)
7 \( 1 - 185. iT - 1.68e4T^{2} \)
11 \( 1 - 574. iT - 1.61e5T^{2} \)
13 \( 1 - 65.8T + 3.71e5T^{2} \)
17 \( 1 - 1.96e3iT - 1.41e6T^{2} \)
19 \( 1 - 611. iT - 2.47e6T^{2} \)
23 \( 1 + 2.97e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.47e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.72e3T + 2.86e7T^{2} \)
37 \( 1 + 7.28e3T + 6.93e7T^{2} \)
41 \( 1 + 6.22e3T + 1.15e8T^{2} \)
43 \( 1 + 1.43e4T + 1.47e8T^{2} \)
47 \( 1 - 5.91e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.06e4T + 4.18e8T^{2} \)
59 \( 1 + 1.24e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.98e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.62e4T + 1.35e9T^{2} \)
71 \( 1 + 5.57e4T + 1.80e9T^{2} \)
73 \( 1 + 3.79e3iT - 2.07e9T^{2} \)
79 \( 1 - 7.40e4T + 3.07e9T^{2} \)
83 \( 1 - 1.17e5T + 3.93e9T^{2} \)
89 \( 1 - 2.81e4T + 5.58e9T^{2} \)
97 \( 1 - 3.89e4iT - 8.58e9T^{2} \)
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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

−11.99101844585291883577638892275, −10.32706161904429318524041890249, −9.670113865997665811749066936242, −8.651364245424109643631983541680, −8.165990347948864355467582653616, −6.31580245522099844779452226328, −4.76439916900224227427283728387, −3.64112106610417110508069542603, −2.39825166664391482985037810403, −1.85392554923913396798296868579, 0.796336946686190108701030226472, 3.07533728115853232899482053389, 3.72498973378529069129259579317, 5.04908833230467166748975692856, 6.78954369425500017572197797618, 7.50626901821373586014936085428, 8.401024585847149342881336398717, 9.178527058111054720622550393708, 10.20645887505308577982058498091, 11.67857067066579992468232810335

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