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Properties

Label 2-200-8.5-c1-0-15
Degree $2$
Conductor $200$
Sign $-0.883 - 0.467i$
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  + (−0.5 − 1.32i)2-s − 2.64i·3-s + (−1.50 + 1.32i)4-s + (−3.50 + 1.32i)6-s − 4·7-s + (2.50 + 1.32i)8-s − 4.00·9-s − 2.64i·11-s + (3.50 + 3.96i)12-s + (2 + 5.29i)14-s + (0.500 − 3.96i)16-s + 3·17-s + (2.00 + 5.29i)18-s − 2.64i·19-s + 10.5i·21-s + (−3.50 + 1.32i)22-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s − 1.52i·3-s + (−0.750 + 0.661i)4-s + (−1.42 + 0.540i)6-s − 1.51·7-s + (0.883 + 0.467i)8-s − 1.33·9-s − 0.797i·11-s + (1.01 + 1.14i)12-s + (0.534 + 1.41i)14-s + (0.125 − 0.992i)16-s + 0.727·17-s + (0.471 + 1.24i)18-s − 0.606i·19-s + 2.30i·21-s + (−0.746 + 0.282i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -0.883 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152453 + 0.614068i\)
\(L(\frac12)\) \(\approx\) \(0.152453 + 0.614068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 1.32i)T \)
5 \( 1 \)
good3 \( 1 + 2.64iT - 3T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 2.64iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2.64iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10.5iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 5.29iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 - 7.93iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 7.93iT - 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−12.27663322469443509525454591162, −11.08447538029037148799030712718, −9.948425133667698485563105980070, −8.937468376288715202210951689621, −7.893476631881274387086974573659, −6.88185987515442619539598265197, −5.79093089168755001397541133467, −3.58908890321402914729492351441, −2.43013874299533290114921261546, −0.62036380948190494547338357103, 3.45433537376310144986067606263, 4.54698847704968478810073318200, 5.73191656978605985197092450600, 6.75150824391477632329555764682, 8.148204716509848239891592748303, 9.347067482071268746374524742045, 9.959362729867815666331769878362, 10.34197819489543942038129863563, 12.05585048460620626423862543456, 13.23784010218841081582418256214

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