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Properties

Label 2-800-40.29-c3-0-28
Degree $2$
Conductor $800$
Sign $0.976 - 0.215i$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
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  + 4.24·3-s − 14.6i·7-s − 8.98·9-s + 14.9i·11-s + 85.6·13-s + 91.9i·17-s + 60.3i·19-s − 62.0i·21-s − 1.33i·23-s − 152.·27-s + 25.5i·29-s − 73.4·31-s + 63.6i·33-s + 211.·37-s + 363.·39-s + ⋯
L(s)  = 1  + 0.816·3-s − 0.789i·7-s − 0.332·9-s + 0.411i·11-s + 1.82·13-s + 1.31i·17-s + 0.728i·19-s − 0.645i·21-s − 0.0121i·23-s − 1.08·27-s + 0.163i·29-s − 0.425·31-s + 0.335i·33-s + 0.938·37-s + 1.49·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ 0.976 - 0.215i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.852863899\)
\(L(\frac12)\) \(\approx\) \(2.852863899\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.24T + 27T^{2} \)
7 \( 1 + 14.6iT - 343T^{2} \)
11 \( 1 - 14.9iT - 1.33e3T^{2} \)
13 \( 1 - 85.6T + 2.19e3T^{2} \)
17 \( 1 - 91.9iT - 4.91e3T^{2} \)
19 \( 1 - 60.3iT - 6.85e3T^{2} \)
23 \( 1 + 1.33iT - 1.21e4T^{2} \)
29 \( 1 - 25.5iT - 2.43e4T^{2} \)
31 \( 1 + 73.4T + 2.97e4T^{2} \)
37 \( 1 - 211.T + 5.06e4T^{2} \)
41 \( 1 - 330.T + 6.89e4T^{2} \)
43 \( 1 - 388.T + 7.95e4T^{2} \)
47 \( 1 + 550. iT - 1.03e5T^{2} \)
53 \( 1 - 187.T + 1.48e5T^{2} \)
59 \( 1 + 779. iT - 2.05e5T^{2} \)
61 \( 1 - 358. iT - 2.26e5T^{2} \)
67 \( 1 - 283.T + 3.00e5T^{2} \)
71 \( 1 - 534.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.11e3T + 4.93e5T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 - 398.T + 7.04e5T^{2} \)
97 \( 1 + 278. iT - 9.12e5T^{2} \)
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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

−9.874960936783325951817240882550, −8.874737606443865736353999781152, −8.277684453274793040466021993758, −7.55033384892606884262163350321, −6.39200285104929878708855007414, −5.62512336395789630184045057701, −3.95375377002164273714242356927, −3.69954288867623208857963516600, −2.22391184181323115585794900279, −1.04363057612402835160083601522, 0.852147414222567461643944553673, 2.42411426983207217746004146903, 3.12012325034798206038224654789, 4.23253982252437111309399684355, 5.58778158409458549129258353086, 6.19198676770696138362176603011, 7.47721216314608341826467514657, 8.295541817978677944048621967800, 9.114894514160649672005360490611, 9.333962744114218066175277046625

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