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Properties

Label 2-825-5.4-c3-0-15
Degree $2$
Conductor $825$
Sign $-0.447 - 0.894i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
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  + 1.56i·2-s − 3i·3-s + 5.56·4-s + 4.68·6-s − 24.2i·7-s + 21.1i·8-s − 9·9-s − 11·11-s − 16.6i·12-s + 84.2i·13-s + 37.8·14-s + 11.4·16-s + 40.9i·17-s − 14.0i·18-s − 120.·19-s + ⋯
L(s)  = 1  + 0.552i·2-s − 0.577i·3-s + 0.695·4-s + 0.318·6-s − 1.31i·7-s + 0.935i·8-s − 0.333·9-s − 0.301·11-s − 0.401i·12-s + 1.79i·13-s + 0.723·14-s + 0.178·16-s + 0.584i·17-s − 0.184i·18-s − 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.417548962\)
\(L(\frac12)\) \(\approx\) \(1.417548962\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 - 1.56iT - 8T^{2} \)
7 \( 1 + 24.2iT - 343T^{2} \)
13 \( 1 - 84.2iT - 2.19e3T^{2} \)
17 \( 1 - 40.9iT - 4.91e3T^{2} \)
19 \( 1 + 120.T + 6.85e3T^{2} \)
23 \( 1 + 9.94iT - 1.21e4T^{2} \)
29 \( 1 + 196.T + 2.43e4T^{2} \)
31 \( 1 - 151.T + 2.97e4T^{2} \)
37 \( 1 - 253. iT - 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
43 \( 1 + 90.4iT - 7.95e4T^{2} \)
47 \( 1 - 483. iT - 1.03e5T^{2} \)
53 \( 1 - 567. iT - 1.48e5T^{2} \)
59 \( 1 - 491.T + 2.05e5T^{2} \)
61 \( 1 - 127.T + 2.26e5T^{2} \)
67 \( 1 + 628. iT - 3.00e5T^{2} \)
71 \( 1 - 309.T + 3.57e5T^{2} \)
73 \( 1 + 1.14e3iT - 3.89e5T^{2} \)
79 \( 1 - 87.4T + 4.93e5T^{2} \)
83 \( 1 - 1.04e3iT - 5.71e5T^{2} \)
89 \( 1 + 390.T + 7.04e5T^{2} \)
97 \( 1 + 165. iT - 9.12e5T^{2} \)
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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

−10.30386421753048524613506961148, −9.065583439570265546043236465074, −8.105198621470469875561563185078, −7.42759581088202130417075880180, −6.59950629561854960321835777380, −6.25856789349670726619429053435, −4.77044576219703593602478050346, −3.82320465095733053031742727084, −2.31043467819412849686108806027, −1.39780908945134306501041338394, 0.33919998085704597988335821108, 2.13539175805934613084495006401, 2.81545229284079861414062612055, 3.80064589808809618090121917482, 5.28290745350518512123847276062, 5.79349038145078469154613908552, 6.91125575860834589365898265952, 8.075647381181435251169386135154, 8.754258902883026697412118107223, 9.861496038629123670084091109164

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