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Properties

Label 2-200-40.3-c1-0-8
Degree $2$
Conductor $200$
Sign $0.854 + 0.519i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 − 0.221i)2-s + (−0.618 − 0.618i)3-s + (1.90 − 0.618i)4-s + (−1 − 0.726i)6-s + (1.90 + 1.90i)7-s + (2.52 − 1.28i)8-s − 2.23i·9-s − 3.23·11-s + (−1.55 − 0.793i)12-s + (−0.726 + 0.726i)13-s + (3.07 + 2.23i)14-s + (3.23 − 2.35i)16-s + (1 − i)17-s + (−0.494 − 3.12i)18-s + 2i·19-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.356 − 0.356i)3-s + (0.951 − 0.309i)4-s + (−0.408 − 0.296i)6-s + (0.718 + 0.718i)7-s + (0.891 − 0.453i)8-s − 0.745i·9-s − 0.975·11-s + (−0.449 − 0.229i)12-s + (−0.201 + 0.201i)13-s + (0.822 + 0.597i)14-s + (0.809 − 0.587i)16-s + (0.242 − 0.242i)17-s + (−0.116 − 0.736i)18-s + 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85183 - 0.518478i\)
\(L(\frac12)\) \(\approx\) \(1.85183 - 0.518478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
5 \( 1 \)
good3 \( 1 + (0.618 + 0.618i)T + 3iT^{2} \)
7 \( 1 + (-1.90 - 1.90i)T + 7iT^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + (0.726 - 0.726i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (4.25 - 4.25i)T - 23iT^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (-0.726 - 0.726i)T + 37iT^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 + (4.61 + 4.61i)T + 43iT^{2} \)
47 \( 1 + (3.35 + 3.35i)T + 47iT^{2} \)
53 \( 1 + (-3.07 + 3.07i)T - 53iT^{2} \)
59 \( 1 - 0.472iT - 59T^{2} \)
61 \( 1 + 0.898iT - 61T^{2} \)
67 \( 1 + (-4.61 + 4.61i)T - 67iT^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (-4.70 - 4.70i)T + 73iT^{2} \)
79 \( 1 - 2.90T + 79T^{2} \)
83 \( 1 + (-6.61 - 6.61i)T + 83iT^{2} \)
89 \( 1 + 2.47iT - 89T^{2} \)
97 \( 1 + (4.23 - 4.23i)T - 97iT^{2} \)
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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

−12.25747138652873382313369264841, −11.76853935599940209942585234955, −10.78971636512886401624567493516, −9.601708707521723670030732019825, −8.098209355612432597993219051274, −7.03846234006223252394823586982, −5.80145738188588626501550726327, −5.10093991843398191310918141067, −3.51414119928654478589056970180, −1.92304072184182163364852366611, 2.34428497374263965322526383139, 4.12672430378085787190694585318, 4.98216671608052652283445951302, 5.96869962013513361005058564487, 7.51669749806652905011847672799, 8.034723415744077122973812645422, 10.03817678258035548536588962602, 10.86568825028244623603151448319, 11.44777664339695106050215223163, 12.76882828377346002043991564719

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