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Properties

Label 2-200-25.6-c1-0-4
Degree $2$
Conductor $200$
Sign $0.968 + 0.248i$
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  + (1.80 − 1.31i)3-s + (−0.690 + 2.12i)5-s + 2.61·7-s + (0.618 − 1.90i)9-s + (0.381 + 1.17i)11-s + (1.04 − 3.21i)13-s + (1.54 + 4.75i)15-s + (−4.23 − 3.07i)17-s + (−6.16 − 4.47i)19-s + (4.73 − 3.44i)21-s + (1.69 + 5.20i)23-s + (−4.04 − 2.93i)25-s + (0.690 + 2.12i)27-s + (−3.54 + 2.57i)29-s + (3.80 + 2.76i)31-s + ⋯
L(s)  = 1  + (1.04 − 0.758i)3-s + (−0.309 + 0.951i)5-s + 0.989·7-s + (0.206 − 0.634i)9-s + (0.115 + 0.354i)11-s + (0.289 − 0.892i)13-s + (0.398 + 1.22i)15-s + (−1.02 − 0.746i)17-s + (−1.41 − 1.02i)19-s + (1.03 − 0.750i)21-s + (0.352 + 1.08i)23-s + (−0.809 − 0.587i)25-s + (0.132 + 0.409i)27-s + (−0.658 + 0.478i)29-s + (0.684 + 0.497i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60034 - 0.202170i\)
\(L(\frac12)\) \(\approx\) \(1.60034 - 0.202170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.690 - 2.12i)T \)
good3 \( 1 + (-1.80 + 1.31i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + (-0.381 - 1.17i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.04 + 3.21i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.23 + 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.16 + 4.47i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.69 - 5.20i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.54 - 2.57i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.80 - 2.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.54 - 7.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.76 + 5.42i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.38T + 43T^{2} \)
47 \( 1 + (0.118 - 0.0857i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.42 - 4.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.736 + 2.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.07 + 12.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (11.0 + 8.05i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-6.97 + 5.06i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.39 - 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.30 + 3.13i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.66 - 2.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.23 - 3.80i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.3 + 8.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.60122340024677383136741084253, −11.29194642353876405929920555030, −10.72996876180955721722164304554, −9.178165556473027426063589102187, −8.230891427521361793792822089441, −7.45746876674645464331543824589, −6.59829440124508259258889095091, −4.79085624826043896341536773143, −3.17370587119169730974811346610, −2.03789560009096921956091206354, 2.02488307079206194451136367777, 4.04559620663862811180051570608, 4.46887371405670977437919867622, 6.16775937932023986386626557286, 7.969011993015351599427181722138, 8.613459789247073535035353261223, 9.168205984833807359921736757614, 10.51486471394432982832110319264, 11.44289945410288717086188345183, 12.57958193139949027027315000715

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