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Properties

Label 2-200-40.29-c5-0-62
Degree $2$
Conductor $200$
Sign $-0.0565 + 0.998i$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.60 + 0.765i)2-s + 17.3·3-s + (30.8 − 8.57i)4-s + (−97.0 + 13.2i)6-s − 9.19i·7-s + (−166. + 71.6i)8-s + 56.8·9-s + 160. i·11-s + (533. − 148. i)12-s − 368.·13-s + (7.03 + 51.5i)14-s + (876. − 528. i)16-s − 1.26e3i·17-s + (−318. + 43.4i)18-s − 2.48e3i·19-s + ⋯
L(s)  = 1  + (−0.990 + 0.135i)2-s + 1.11·3-s + (0.963 − 0.268i)4-s + (−1.10 + 0.150i)6-s − 0.0708i·7-s + (−0.918 + 0.395i)8-s + 0.233·9-s + 0.399i·11-s + (1.07 − 0.297i)12-s − 0.604·13-s + (0.00958 + 0.0702i)14-s + (0.856 − 0.516i)16-s − 1.05i·17-s + (−0.231 + 0.0316i)18-s − 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0565 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0565 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.0565 + 0.998i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -0.0565 + 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.225915788\)
\(L(\frac12)\) \(\approx\) \(1.225915788\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.60 - 0.765i)T \)
5 \( 1 \)
good3 \( 1 - 17.3T + 243T^{2} \)
7 \( 1 + 9.19iT - 1.68e4T^{2} \)
11 \( 1 - 160. iT - 1.61e5T^{2} \)
13 \( 1 + 368.T + 3.71e5T^{2} \)
17 \( 1 + 1.26e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.48e3iT - 2.47e6T^{2} \)
23 \( 1 - 422. iT - 6.43e6T^{2} \)
29 \( 1 + 5.66e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.38e3T + 2.86e7T^{2} \)
37 \( 1 + 3.56e3T + 6.93e7T^{2} \)
41 \( 1 + 5.94e3T + 1.15e8T^{2} \)
43 \( 1 + 1.06e4T + 1.47e8T^{2} \)
47 \( 1 - 9.24e3iT - 2.29e8T^{2} \)
53 \( 1 + 8.97e3T + 4.18e8T^{2} \)
59 \( 1 + 2.74e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.05e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.96e3T + 1.35e9T^{2} \)
71 \( 1 - 6.72e4T + 1.80e9T^{2} \)
73 \( 1 + 8.57e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.65e4T + 3.07e9T^{2} \)
83 \( 1 - 3.02e4T + 3.93e9T^{2} \)
89 \( 1 + 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.38e5iT - 8.58e9T^{2} \)
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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

−11.21674490758310487239033165469, −9.871762792024778912064414171280, −9.348925237016177579488179675499, −8.360697776758855776852228950804, −7.52836209191405280782951975617, −6.57868797251436501497605356574, −4.89300183238820448491923779379, −3.02627148606664269833690788294, −2.19104059147723937472580927974, −0.43032889807148529660421991648, 1.46682099910730364225002438531, 2.68415318251682251563530448994, 3.74114700066589416693045636734, 5.80917321002664058443489174106, 7.08864363342897289036343964055, 8.300982659859572060721508250131, 8.539249639978777502820053310504, 9.819217764058962127134790954003, 10.47487686935876802408947811017, 11.77947578501420200223326241220

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