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Properties

Label 2-200-40.29-c5-0-3
Degree $2$
Conductor $200$
Sign $0.340 - 0.940i$
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  + (−3.01 − 4.78i)2-s − 25.4·3-s + (−13.7 + 28.8i)4-s + (76.7 + 121. i)6-s − 56.4i·7-s + (179. − 21.0i)8-s + 403.·9-s + 261. i·11-s + (350. − 734. i)12-s − 720.·13-s + (−270. + 170. i)14-s + (−643. − 796. i)16-s − 1.87e3i·17-s + (−1.21e3 − 1.93e3i)18-s − 1.99e3i·19-s + ⋯
L(s)  = 1  + (−0.533 − 0.845i)2-s − 1.63·3-s + (−0.431 + 0.902i)4-s + (0.870 + 1.38i)6-s − 0.435i·7-s + (0.993 − 0.116i)8-s + 1.66·9-s + 0.650i·11-s + (0.703 − 1.47i)12-s − 1.18·13-s + (−0.368 + 0.232i)14-s + (−0.628 − 0.778i)16-s − 1.57i·17-s + (−0.886 − 1.40i)18-s − 1.26i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.340 - 0.940i$
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: \((2,\ 200,\ (\ :5/2),\ 0.340 - 0.940i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.06449559710\)
\(L(\frac12)\) \(\approx\) \(0.06449559710\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.01 + 4.78i)T \)
5 \( 1 \)
good3 \( 1 + 25.4T + 243T^{2} \)
7 \( 1 + 56.4iT - 1.68e4T^{2} \)
11 \( 1 - 261. iT - 1.61e5T^{2} \)
13 \( 1 + 720.T + 3.71e5T^{2} \)
17 \( 1 + 1.87e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.99e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.57e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.70e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.73e3T + 2.86e7T^{2} \)
37 \( 1 - 1.22e4T + 6.93e7T^{2} \)
41 \( 1 - 1.49e4T + 1.15e8T^{2} \)
43 \( 1 + 1.81e4T + 1.47e8T^{2} \)
47 \( 1 - 2.14e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.60e3T + 4.18e8T^{2} \)
59 \( 1 - 2.68e3iT - 7.14e8T^{2} \)
61 \( 1 + 4.45e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.24e4T + 1.35e9T^{2} \)
71 \( 1 - 8.18e3T + 1.80e9T^{2} \)
73 \( 1 - 4.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.63e4T + 3.07e9T^{2} \)
83 \( 1 + 6.16e4T + 3.93e9T^{2} \)
89 \( 1 + 5.32e4T + 5.58e9T^{2} \)
97 \( 1 + 3.92e4iT - 8.58e9T^{2} \)
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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

−11.55435043957096185855263987412, −10.98060769564489645095924440641, −10.00800744166039009097320537657, −9.265560042984548662307482510066, −7.45513134961313368982068990654, −6.86655559169714322489510993428, −5.09607176677537199474963108361, −4.46334634225894755128299636275, −2.51442701447673796191292707058, −0.793347117645819083215733546851, 0.04479178163330142892008756941, 1.52476739418804202298298708713, 4.25385390246494519845124699797, 5.73275175679603373649520249230, 5.78954556810555862216476036172, 7.14017390923063984741051358360, 8.168296200122843855581897396755, 9.542965566334346530953841950031, 10.39446089234837862077892882583, 11.23421340443016382909258318151

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