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Properties

Label 2-200-8.5-c5-0-83
Degree $2$
Conductor $200$
Sign $-0.587 + 0.809i$
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  + (5.37 − 1.74i)2-s + 25.2i·3-s + (25.8 − 18.8i)4-s + (44.2 + 135. i)6-s − 185.·7-s + (106. − 146. i)8-s − 395.·9-s − 574. i·11-s + (475. + 653. i)12-s + 65.8i·13-s + (−996. + 324. i)14-s + (315. − 974. i)16-s − 1.96e3·17-s + (−2.12e3 + 691. i)18-s + 611. i·19-s + ⋯
L(s)  = 1  + (0.950 − 0.309i)2-s + 1.62i·3-s + (0.808 − 0.588i)4-s + (0.501 + 1.54i)6-s − 1.42·7-s + (0.587 − 0.809i)8-s − 1.62·9-s − 1.43i·11-s + (0.953 + 1.31i)12-s + 0.108i·13-s + (−1.35 + 0.441i)14-s + (0.307 − 0.951i)16-s − 1.65·17-s + (−1.54 + 0.503i)18-s + 0.388i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5455025990\)
\(L(\frac12)\) \(\approx\) \(0.5455025990\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.37 + 1.74i)T \)
5 \( 1 \)
good3 \( 1 - 25.2iT - 243T^{2} \)
7 \( 1 + 185.T + 1.68e4T^{2} \)
11 \( 1 + 574. iT - 1.61e5T^{2} \)
13 \( 1 - 65.8iT - 3.71e5T^{2} \)
17 \( 1 + 1.96e3T + 1.41e6T^{2} \)
19 \( 1 - 611. iT - 2.47e6T^{2} \)
23 \( 1 + 2.97e3T + 6.43e6T^{2} \)
29 \( 1 + 4.47e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.72e3T + 2.86e7T^{2} \)
37 \( 1 - 7.28e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.22e3T + 1.15e8T^{2} \)
43 \( 1 + 1.43e4iT - 1.47e8T^{2} \)
47 \( 1 + 5.91e3T + 2.29e8T^{2} \)
53 \( 1 - 2.06e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.24e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.98e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.62e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.57e4T + 1.80e9T^{2} \)
73 \( 1 + 3.79e3T + 2.07e9T^{2} \)
79 \( 1 + 7.40e4T + 3.07e9T^{2} \)
83 \( 1 - 1.17e5iT - 3.93e9T^{2} \)
89 \( 1 + 2.81e4T + 5.58e9T^{2} \)
97 \( 1 + 3.89e4T + 8.58e9T^{2} \)
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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

−11.16566546652182031090775644345, −10.30474856053899167453867603786, −9.660089446592179527768478561741, −8.544916449470713103009426508917, −6.48511595744016032938198014197, −5.82813466962476150032659723243, −4.46030077258892345403514228820, −3.65017613315218931055528681769, −2.73991162484245898978468474023, −0.10635196841686975060604406894, 1.88601000764777110741245062897, 2.85007069012564550219533026076, 4.45710804132106350074308541181, 6.06883935106683877859980455972, 6.76030808836380826653684984227, 7.30332253349314412471542053193, 8.567261034371623684359026245923, 10.02955541936144796536500280271, 11.50860994150093205131282135235, 12.35916907574727897019251994269

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