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Properties

Label 2-200-40.3-c1-0-13
Degree $2$
Conductor $200$
Sign $-0.973 - 0.229i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−2 − 2i)3-s − 2i·4-s + 4·6-s + (2 + 2i)8-s + 5i·9-s − 6·11-s + (−4 + 4i)12-s − 4·16-s + (−4 + 4i)17-s + (−5 − 5i)18-s + 2i·19-s + (6 − 6i)22-s − 8i·24-s + (4 − 4i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.15 − 1.15i)3-s − i·4-s + 1.63·6-s + (0.707 + 0.707i)8-s + 1.66i·9-s − 1.80·11-s + (−1.15 + 1.15i)12-s − 16-s + (−0.970 + 0.970i)17-s + (−1.17 − 1.17i)18-s + 0.458i·19-s + (1.27 − 1.27i)22-s − 1.63i·24-s + (0.769 − 0.769i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
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: \(1\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 2i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (4 - 4i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
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 \( 1 + (-6 + 6i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12 + 12i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (2 + 2i)T + 83iT^{2} \)
89 \( 1 + 18iT - 89T^{2} \)
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

−11.86543843523725153069004658553, −10.79435693539942004941676460471, −10.25028305894285021328916407748, −8.536829961717031401451695817128, −7.71703435992756767366116558877, −6.80108311468957659426418931131, −5.86748235504678641306859102020, −5.01538615927079748835865910659, −1.94465861321050862254360420318, 0, 2.82311213713830116792009835201, 4.44058643613850798526754055740, 5.31096702673219580020982660690, 6.90045596591976007798566833346, 8.239392002889202088141311778017, 9.468078530657130202143325614055, 10.19208672449491571158597493896, 10.99296282352174022324289171520, 11.51002869178452917126363312877

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