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Properties

Label 2-200-40.27-c3-0-11
Degree $2$
Conductor $200$
Sign $-0.0955 - 0.995i$
Analytic cond. $11.8003$
Root an. cond. $3.43516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.76 + 0.607i)2-s + (2.02 − 2.02i)3-s + (7.26 − 3.35i)4-s + (−4.36 + 6.83i)6-s + (1.63 − 1.63i)7-s + (−18.0 + 13.6i)8-s + 18.7i·9-s − 62.3·11-s + (7.91 − 21.5i)12-s + (47.3 + 47.3i)13-s + (−3.51 + 5.50i)14-s + (41.4 − 48.7i)16-s + (−26.6 − 26.6i)17-s + (−11.4 − 51.8i)18-s + 59.7i·19-s + ⋯
L(s)  = 1  + (−0.976 + 0.214i)2-s + (0.390 − 0.390i)3-s + (0.907 − 0.419i)4-s + (−0.297 + 0.464i)6-s + (0.0881 − 0.0881i)7-s + (−0.796 + 0.604i)8-s + 0.695i·9-s − 1.71·11-s + (0.190 − 0.517i)12-s + (1.01 + 1.01i)13-s + (−0.0671 + 0.105i)14-s + (0.648 − 0.761i)16-s + (−0.379 − 0.379i)17-s + (−0.149 − 0.679i)18-s + 0.721i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0955 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0955 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.0955 - 0.995i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ -0.0955 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.573868 + 0.631618i\)
\(L(\frac12)\) \(\approx\) \(0.573868 + 0.631618i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.76 - 0.607i)T \)
5 \( 1 \)
good3 \( 1 + (-2.02 + 2.02i)T - 27iT^{2} \)
7 \( 1 + (-1.63 + 1.63i)T - 343iT^{2} \)
11 \( 1 + 62.3T + 1.33e3T^{2} \)
13 \( 1 + (-47.3 - 47.3i)T + 2.19e3iT^{2} \)
17 \( 1 + (26.6 + 26.6i)T + 4.91e3iT^{2} \)
19 \( 1 - 59.7iT - 6.85e3T^{2} \)
23 \( 1 + (-6.56 - 6.56i)T + 1.21e4iT^{2} \)
29 \( 1 - 73.0T + 2.43e4T^{2} \)
31 \( 1 - 168. iT - 2.97e4T^{2} \)
37 \( 1 + (97.0 - 97.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 27.5T + 6.89e4T^{2} \)
43 \( 1 + (204. - 204. i)T - 7.95e4iT^{2} \)
47 \( 1 + (165. - 165. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-372. - 372. i)T + 1.48e5iT^{2} \)
59 \( 1 - 67.1iT - 2.05e5T^{2} \)
61 \( 1 - 659. iT - 2.26e5T^{2} \)
67 \( 1 + (506. + 506. i)T + 3.00e5iT^{2} \)
71 \( 1 + 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + (-685. + 685. i)T - 3.89e5iT^{2} \)
79 \( 1 - 751.T + 4.93e5T^{2} \)
83 \( 1 + (251. - 251. i)T - 5.71e5iT^{2} \)
89 \( 1 + 497. iT - 7.04e5T^{2} \)
97 \( 1 + (305. + 305. i)T + 9.12e5iT^{2} \)
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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

−12.12710783052003900392320172308, −10.91625649003424388942943366294, −10.40237107499877096994631827180, −9.068852296792440101525192278623, −8.182929968414229425484284470414, −7.50268750117056927180320224741, −6.33373967355236108846192715793, −4.99419937575233181256493297110, −2.84355134422789113013789152057, −1.59041916230596748163872511662, 0.47295625486868143165113227112, 2.48916864206866596518650711738, 3.61684751897356072023352411708, 5.49651791187405477612964876242, 6.80276808208182693233059645426, 8.131097622849540758992034757580, 8.625121134246960058579658673267, 9.859040240348009987336069383599, 10.54557051721895855093195604363, 11.42404897437438211358093520412

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