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Properties

Label 2-20e2-5.4-c3-0-17
Degree $2$
Conductor $400$
Sign $0.894 + 0.447i$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
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  + 5i·3-s − 2i·7-s + 2·9-s − 39·11-s − 84i·13-s − 61i·17-s + 151·19-s + 10·21-s − 58i·23-s + 145i·27-s − 192·29-s + 18·31-s − 195i·33-s − 138i·37-s + 420·39-s + ⋯
L(s)  = 1  + 0.962i·3-s − 0.107i·7-s + 0.0740·9-s − 1.06·11-s − 1.79i·13-s − 0.870i·17-s + 1.82·19-s + 0.103·21-s − 0.525i·23-s + 1.03i·27-s − 1.22·29-s + 0.104·31-s − 1.02i·33-s − 0.613i·37-s + 1.72·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.641647825\)
\(L(\frac12)\) \(\approx\) \(1.641647825\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 5iT - 27T^{2} \)
7 \( 1 + 2iT - 343T^{2} \)
11 \( 1 + 39T + 1.33e3T^{2} \)
13 \( 1 + 84iT - 2.19e3T^{2} \)
17 \( 1 + 61iT - 4.91e3T^{2} \)
19 \( 1 - 151T + 6.85e3T^{2} \)
23 \( 1 + 58iT - 1.21e4T^{2} \)
29 \( 1 + 192T + 2.43e4T^{2} \)
31 \( 1 - 18T + 2.97e4T^{2} \)
37 \( 1 + 138iT - 5.06e4T^{2} \)
41 \( 1 - 229T + 6.89e4T^{2} \)
43 \( 1 + 164iT - 7.95e4T^{2} \)
47 \( 1 - 212iT - 1.03e5T^{2} \)
53 \( 1 + 578iT - 1.48e5T^{2} \)
59 \( 1 + 336T + 2.05e5T^{2} \)
61 \( 1 - 858T + 2.26e5T^{2} \)
67 \( 1 - 209iT - 3.00e5T^{2} \)
71 \( 1 - 780T + 3.57e5T^{2} \)
73 \( 1 - 403iT - 3.89e5T^{2} \)
79 \( 1 + 230T + 4.93e5T^{2} \)
83 \( 1 + 1.29e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 - 382iT - 9.12e5T^{2} \)
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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

−10.58128241432074999638222300701, −9.990878595612598660703277764377, −9.209445740719670952106610329777, −7.910282377596835729813208521935, −7.28530267314271904419522932127, −5.50418318458000514619121336293, −5.12173810887201299868240399204, −3.70185621237209127973616967117, −2.73829305316966324934325726136, −0.60961234309670321078894777049, 1.26339501294110320429213289339, 2.33795150742758746190031377900, 3.88474608032378130037057038908, 5.21753486656324671773540152447, 6.30567275052962288945521705611, 7.29851259318978972731056174477, 7.85133730311744519298889962255, 9.103346589423949317988264741806, 9.918054300281964762245625564133, 11.12465080072562340770867535757

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