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Properties

Label 2-200-40.27-c1-0-1
Degree $2$
Conductor $200$
Sign $-0.996 - 0.0822i$
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  + (0.221 + 1.39i)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 + (−1.28 − 2.52i)8-s + 2.23i·9-s − 3.23·11-s + (0.793 − 1.55i)12-s + (0.726 + 0.726i)13-s + (−3.07 − 2.23i)14-s + (3.23 − 2.35i)16-s + (1 + i)17-s + (−3.12 + 0.494i)18-s − 2i·19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)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 + (−0.453 − 0.891i)8-s + 0.745i·9-s − 0.975·11-s + (0.229 − 0.449i)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 + (−0.736 + 0.116i)18-s − 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.996 - 0.0822i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -0.996 - 0.0822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0318701 + 0.773717i\)
\(L(\frac12)\) \(\approx\) \(0.0318701 + 0.773717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
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

−13.09854571297799241539362320015, −12.21426425805829093162963781611, −10.85208955301479158904048629756, −9.855190046994486125334330885316, −8.839033445629448769823154725856, −7.83996766339985152216532702506, −6.66032761174620370535655451933, −5.54194311199417550720019623236, −4.79123454200317900956382310572, −3.07775458957219750842948249413, 0.68364794438649643120869448681, 2.83191318942971110173199148259, 4.07078090330073732339788102932, 5.51400583549458295656811837794, 6.68196395538748969063637287406, 8.042605738273087300759863537376, 9.339938272313211272628096600651, 10.21961847863987285267993808865, 10.99759944351771406297747980069, 12.11582204856619076256902012826

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