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Properties

Label 2-200-40.3-c1-0-15
Degree $2$
Conductor $200$
Sign $-0.991 + 0.130i$
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 − i)2-s + (−2.22 − 2.22i)3-s − 2i·4-s − 4.44·6-s + (−2 − 2i)8-s + 6.89i·9-s + 0.550·11-s + (−4.44 + 4.44i)12-s − 4·16-s + (1.67 − 1.67i)17-s + (6.89 + 6.89i)18-s − 8.34i·19-s + (0.550 − 0.550i)22-s + 8.89i·24-s + (8.67 − 8.67i)27-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−1.28 − 1.28i)3-s − i·4-s − 1.81·6-s + (−0.707 − 0.707i)8-s + 2.29i·9-s + 0.165·11-s + (−1.28 + 1.28i)12-s − 16-s + (0.406 − 0.406i)17-s + (1.62 + 1.62i)18-s − 1.91i·19-s + (0.117 − 0.117i)22-s + 1.81i·24-s + (1.66 − 1.66i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.991 + 0.130i$
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: \((2,\ 200,\ (\ :1/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0674611 - 1.02732i\)
\(L(\frac12)\) \(\approx\) \(0.0674611 - 1.02732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 \)
good3 \( 1 + (2.22 + 2.22i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 0.550T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \)
19 \( 1 + 8.34iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 12.7T + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (5.57 - 5.57i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (7.22 + 7.22i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 + (-12 + 12i)T - 97iT^{2} \)
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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

−11.97334605219536658353673093852, −11.34092940492418680626871772698, −10.61963208972812507735020359237, −9.222388585541298374123273188096, −7.47121923904750975706704075665, −6.57068787004963010496849724737, −5.62786658191637446009706905229, −4.62490756329676464249240048063, −2.52182002874068447298479444649, −0.879868047995396952604262772193, 3.61167382106582127343927709295, 4.46334661418564516802034164156, 5.67084716115985473628009867567, 6.16520275082593948679464711423, 7.66520205942067832575048690576, 9.063529133533466490892419008060, 10.15081175199595013384971987651, 11.06631068167425087423492015440, 12.06320042319062814062143586895, 12.63065183198354886349108728080

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