Properties

Label 2-825-5.4-c3-0-27
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  − 0.368i·2-s − 3i·3-s + 7.86·4-s − 1.10·6-s + 26.5i·7-s − 5.84i·8-s − 9·9-s − 11·11-s − 23.5i·12-s + 50.4i·13-s + 9.79·14-s + 60.7·16-s − 108. i·17-s + 3.31i·18-s − 19.1·19-s + ⋯
L(s)  = 1  − 0.130i·2-s − 0.577i·3-s + 0.983·4-s − 0.0752·6-s + 1.43i·7-s − 0.258i·8-s − 0.333·9-s − 0.301·11-s − 0.567i·12-s + 1.07i·13-s + 0.187·14-s + 0.949·16-s − 1.55i·17-s + 0.0434i·18-s − 0.230·19-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

\(L(2)\) \(\approx\) \(2.108745618\)
\(L(\frac12)\) \(\approx\) \(2.108745618\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.368iT - 8T^{2} \)
7 \( 1 - 26.5iT - 343T^{2} \)
13 \( 1 - 50.4iT - 2.19e3T^{2} \)
17 \( 1 + 108. iT - 4.91e3T^{2} \)
19 \( 1 + 19.1T + 6.85e3T^{2} \)
23 \( 1 - 60.4iT - 1.21e4T^{2} \)
29 \( 1 - 39.2T + 2.43e4T^{2} \)
31 \( 1 + 22.4T + 2.97e4T^{2} \)
37 \( 1 - 345. iT - 5.06e4T^{2} \)
41 \( 1 + 96.3T + 6.89e4T^{2} \)
43 \( 1 - 335. iT - 7.95e4T^{2} \)
47 \( 1 - 514. iT - 1.03e5T^{2} \)
53 \( 1 - 131. iT - 1.48e5T^{2} \)
59 \( 1 - 210.T + 2.05e5T^{2} \)
61 \( 1 + 68.9T + 2.26e5T^{2} \)
67 \( 1 + 202. iT - 3.00e5T^{2} \)
71 \( 1 + 645.T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3iT - 3.89e5T^{2} \)
79 \( 1 + 321.T + 4.93e5T^{2} \)
83 \( 1 - 840. iT - 5.71e5T^{2} \)
89 \( 1 - 1.44e3T + 7.04e5T^{2} \)
97 \( 1 + 1.60e3iT - 9.12e5T^{2} \)
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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

−9.879079273375309720468396065364, −9.157933324481338953438185400572, −8.223694649638561045284602643230, −7.31116894609676132339522104673, −6.53533869598285758802552023819, −5.79157081449370786443426788361, −4.78030669141799916228647644311, −3.03075039422674944566925887642, −2.41987805229767001914870497177, −1.37493772962213065366481312498, 0.52323288023135620452585499532, 1.98055114899859509412578691746, 3.33436736976202426069314495849, 4.06074517734262523009569838503, 5.33255494919378980216958158151, 6.21527988574151812093831285457, 7.15876773866656233253628564722, 7.88068120090430820094693480274, 8.687325678173371746881781246170, 10.24508407955545115441812222214

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