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Properties

Label 2-200-40.27-c1-0-14
Degree $2$
Conductor $200$
Sign $0.858 + 0.512i$
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  + (1.40 − 0.178i)2-s + (1.22 − 1.22i)3-s + (1.93 − 0.5i)4-s + (1.5 − 1.93i)6-s + (−3.16 + 3.16i)7-s + (2.62 − 1.04i)8-s − 11-s + (1.75 − 2.98i)12-s + (−3.16 − 3.16i)13-s + (−3.87 + 5.00i)14-s + (3.50 − 1.93i)16-s + (−3.67 − 3.67i)17-s + 3i·19-s + 7.74i·21-s + (−1.40 + 0.178i)22-s + ⋯
L(s)  = 1  + (0.992 − 0.126i)2-s + (0.707 − 0.707i)3-s + (0.968 − 0.250i)4-s + (0.612 − 0.790i)6-s + (−1.19 + 1.19i)7-s + (0.929 − 0.370i)8-s − 0.301·11-s + (0.507 − 0.861i)12-s + (−0.877 − 0.877i)13-s + (−1.03 + 1.33i)14-s + (0.875 − 0.484i)16-s + (−0.891 − 0.891i)17-s + 0.688i·19-s + 1.69i·21-s + (−0.299 + 0.0379i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.858 + 0.512i$
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: \((2,\ 200,\ (\ :1/2),\ 0.858 + 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17932 - 0.600240i\)
\(L(\frac12)\) \(\approx\) \(2.17932 - 0.600240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.178i)T \)
5 \( 1 \)
good3 \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \)
7 \( 1 + (3.16 - 3.16i)T - 7iT^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (3.16 + 3.16i)T + 13iT^{2} \)
17 \( 1 + (3.67 + 3.67i)T + 17iT^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-3.16 + 3.16i)T - 37iT^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + (2.44 - 2.44i)T - 43iT^{2} \)
47 \( 1 + (3.16 - 3.16i)T - 47iT^{2} \)
53 \( 1 + (-6.32 - 6.32i)T + 53iT^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 7.74iT - 61T^{2} \)
67 \( 1 + (3.67 + 3.67i)T + 67iT^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - 73iT^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \)
89 \( 1 - 13iT - 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−12.58950752396516733248942919084, −11.92934263515974397767279004539, −10.50459443753920906741155098202, −9.443822255977453127317464283671, −8.161943344752514635737680501413, −7.10220863963125168165984060409, −6.07807075330538878324229002649, −4.94437029922387267978666672080, −3.02363864448979790835326015525, −2.42096631410509437341543282928, 2.73162574121744782582008007711, 3.86930602517432593438216884136, 4.62438911605500284573794078914, 6.45751130990050597610779454448, 7.07908254170451811452280665472, 8.560167765604545541800233202844, 9.844483970474953072377077791693, 10.44461326063025757296575645191, 11.72235782351489676001174455887, 12.86831858552974150233384347489

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