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Properties

Label 2-200-40.29-c3-0-44
Degree $2$
Conductor $200$
Sign $0.581 + 0.813i$
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  + (2.64 − i)2-s + 5.29·3-s + (6.00 − 5.29i)4-s + (14.0 − 5.29i)6-s − 8i·7-s + (10.5 − 20.0i)8-s + 1.00·9-s − 15.8i·11-s + (31.7 − 28.0i)12-s + 52.9·13-s + (−8 − 21.1i)14-s + (8.00 − 63.4i)16-s − 14i·17-s + (2.64 − 1.00i)18-s + 37.0i·19-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + 1.01·3-s + (0.750 − 0.661i)4-s + (0.952 − 0.360i)6-s − 0.431i·7-s + (0.467 − 0.883i)8-s + 0.0370·9-s − 0.435i·11-s + (0.763 − 0.673i)12-s + 1.12·13-s + (−0.152 − 0.404i)14-s + (0.125 − 0.992i)16-s − 0.199i·17-s + (0.0346 − 0.0130i)18-s + 0.447i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.72539 - 1.91667i\)
\(L(\frac12)\) \(\approx\) \(3.72539 - 1.91667i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.64 + i)T \)
5 \( 1 \)
good3 \( 1 - 5.29T + 27T^{2} \)
7 \( 1 + 8iT - 343T^{2} \)
11 \( 1 + 15.8iT - 1.33e3T^{2} \)
13 \( 1 - 52.9T + 2.19e3T^{2} \)
17 \( 1 + 14iT - 4.91e3T^{2} \)
19 \( 1 - 37.0iT - 6.85e3T^{2} \)
23 \( 1 - 152iT - 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 - 224T + 2.97e4T^{2} \)
37 \( 1 + 243.T + 5.06e4T^{2} \)
41 \( 1 + 70T + 6.89e4T^{2} \)
43 \( 1 + 439.T + 7.95e4T^{2} \)
47 \( 1 - 336iT - 1.03e5T^{2} \)
53 \( 1 - 31.7T + 1.48e5T^{2} \)
59 \( 1 + 534. iT - 2.05e5T^{2} \)
61 \( 1 - 95.2iT - 2.26e5T^{2} \)
67 \( 1 + 174.T + 3.00e5T^{2} \)
71 \( 1 + 72T + 3.57e5T^{2} \)
73 \( 1 - 294iT - 3.89e5T^{2} \)
79 \( 1 - 464T + 4.93e5T^{2} \)
83 \( 1 + 545.T + 5.71e5T^{2} \)
89 \( 1 + 266T + 7.04e5T^{2} \)
97 \( 1 - 994iT - 9.12e5T^{2} \)
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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

−11.89811872622107508611177990326, −11.04099556576259945823389181165, −10.02604811450487725747552045082, −8.855734232509359083120939632205, −7.80269237312966961466479748827, −6.54419702812773817185770952003, −5.34999267387999694658590908998, −3.81068186683113926582392619622, −3.09197810986066512827418445773, −1.48044064865723989614555377506, 2.17600841692856233911964188014, 3.27595981300350026494698688499, 4.47148527054231132169582418371, 5.86418345420620279456466499598, 6.90809789943188335012844106266, 8.265231492652636157862774087866, 8.701030867503481429733882334268, 10.24159623444617435314075404217, 11.47151008806628165366020026578, 12.34140438840381398196082760737

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