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Properties

Label 2-200-40.27-c3-0-8
Degree $2$
Conductor $200$
Sign $-0.985 - 0.169i$
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  + (−2.24 + 1.71i)2-s + (−3.49 + 3.49i)3-s + (2.11 − 7.71i)4-s + (1.86 − 13.8i)6-s + (4.97 − 4.97i)7-s + (8.47 + 20.9i)8-s + 2.63i·9-s + 29.8·11-s + (19.5 + 34.3i)12-s + (−13.5 − 13.5i)13-s + (−2.65 + 19.7i)14-s + (−55.0 − 32.6i)16-s + (61.7 + 61.7i)17-s + (−4.52 − 5.93i)18-s + 131. i·19-s + ⋯
L(s)  = 1  + (−0.795 + 0.606i)2-s + (−0.671 + 0.671i)3-s + (0.264 − 0.964i)4-s + (0.126 − 0.941i)6-s + (0.268 − 0.268i)7-s + (0.374 + 0.927i)8-s + 0.0977i·9-s + 0.817·11-s + (0.470 + 0.825i)12-s + (−0.289 − 0.289i)13-s + (−0.0506 + 0.376i)14-s + (−0.859 − 0.510i)16-s + (0.881 + 0.881i)17-s + (−0.0592 − 0.0777i)18-s + 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.985 - 0.169i$
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: \((2,\ 200,\ (\ :3/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0530822 + 0.623636i\)
\(L(\frac12)\) \(\approx\) \(0.0530822 + 0.623636i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.24 - 1.71i)T \)
5 \( 1 \)
good3 \( 1 + (3.49 - 3.49i)T - 27iT^{2} \)
7 \( 1 + (-4.97 + 4.97i)T - 343iT^{2} \)
11 \( 1 - 29.8T + 1.33e3T^{2} \)
13 \( 1 + (13.5 + 13.5i)T + 2.19e3iT^{2} \)
17 \( 1 + (-61.7 - 61.7i)T + 4.91e3iT^{2} \)
19 \( 1 - 131. iT - 6.85e3T^{2} \)
23 \( 1 + (-1.13 - 1.13i)T + 1.21e4iT^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 + 276. iT - 2.97e4T^{2} \)
37 \( 1 + (278. - 278. i)T - 5.06e4iT^{2} \)
41 \( 1 + 225.T + 6.89e4T^{2} \)
43 \( 1 + (171. - 171. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-86.1 + 86.1i)T - 1.03e5iT^{2} \)
53 \( 1 + (-108. - 108. i)T + 1.48e5iT^{2} \)
59 \( 1 + 157. iT - 2.05e5T^{2} \)
61 \( 1 - 791. iT - 2.26e5T^{2} \)
67 \( 1 + (-3.80 - 3.80i)T + 3.00e5iT^{2} \)
71 \( 1 - 58.6iT - 3.57e5T^{2} \)
73 \( 1 + (452. - 452. i)T - 3.89e5iT^{2} \)
79 \( 1 - 821.T + 4.93e5T^{2} \)
83 \( 1 + (512. - 512. i)T - 5.71e5iT^{2} \)
89 \( 1 - 500. iT - 7.04e5T^{2} \)
97 \( 1 + (-60.7 - 60.7i)T + 9.12e5iT^{2} \)
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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

−12.10865352110777590253084993946, −11.22160605624139703621827637835, −10.25046447496972062844601612690, −9.771548557525506419864749398481, −8.369894817944985347318464578151, −7.54034154029358346433787274321, −6.13921323436062507814060872765, −5.36657718564362270444084526283, −4.00823103549476701456002131637, −1.53771751899638174044668450889, 0.40208433718746643173992728513, 1.75370889609788078169522801314, 3.41982400650058939150844831967, 5.16058584656271462205250825048, 6.75681868031072221142685223904, 7.29167851044599578554449885961, 8.789015809365628788834355142658, 9.451652948643270242666265468140, 10.75549631875032621571405493452, 11.72582335702803278707717944283

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