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Properties

Label 2-825-5.4-c3-0-17
Degree $2$
Conductor $825$
Sign $-0.894 - 0.447i$
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  + 3.37i·2-s − 3i·3-s − 3.37·4-s + 10.1·6-s − 4.74i·7-s + 15.6i·8-s − 9·9-s + 11·11-s + 10.1i·12-s + 15.0i·13-s + 16·14-s − 79.6·16-s + 73.1i·17-s − 30.3i·18-s + 78.7·19-s + ⋯
L(s)  = 1  + 1.19i·2-s − 0.577i·3-s − 0.421·4-s + 0.688·6-s − 0.256i·7-s + 0.689i·8-s − 0.333·9-s + 0.301·11-s + 0.243i·12-s + 0.320i·13-s + 0.305·14-s − 1.24·16-s + 1.04i·17-s − 0.397i·18-s + 0.950·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 825,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.635319947\)
\(L(\frac12)\) \(\approx\) \(1.635319947\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.37iT - 8T^{2} \)
7 \( 1 + 4.74iT - 343T^{2} \)
13 \( 1 - 15.0iT - 2.19e3T^{2} \)
17 \( 1 - 73.1iT - 4.91e3T^{2} \)
19 \( 1 - 78.7T + 6.85e3T^{2} \)
23 \( 1 + 112iT - 1.21e4T^{2} \)
29 \( 1 + 243.T + 2.43e4T^{2} \)
31 \( 1 - 278.T + 2.97e4T^{2} \)
37 \( 1 - 102. iT - 5.06e4T^{2} \)
41 \( 1 + 241.T + 6.89e4T^{2} \)
43 \( 1 - 280. iT - 7.95e4T^{2} \)
47 \( 1 + 169. iT - 1.03e5T^{2} \)
53 \( 1 - 409. iT - 1.48e5T^{2} \)
59 \( 1 + 196T + 2.05e5T^{2} \)
61 \( 1 + 701.T + 2.26e5T^{2} \)
67 \( 1 - 900. iT - 3.00e5T^{2} \)
71 \( 1 - 756.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 327.T + 4.93e5T^{2} \)
83 \( 1 - 756. iT - 5.71e5T^{2} \)
89 \( 1 + 508.T + 7.04e5T^{2} \)
97 \( 1 - 614. iT - 9.12e5T^{2} \)
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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

−10.08626953315189299957917173154, −8.985463968885444172788613708195, −8.227709112789237858780075053400, −7.54209529625799401250353981242, −6.70884545013897300925619159669, −6.13239770811674301740016979319, −5.18784400265825709432963381973, −4.05685286102809497008716435209, −2.59743699604936556913094328077, −1.29115644146394651132330971686, 0.43933794673993289051398098611, 1.76535335782340461396077217258, 2.96504690377745871060080933498, 3.62304615368402697042245789177, 4.76844105866890612281669219017, 5.70820420761944387017170179144, 6.92193925396696083658119194414, 7.86261677784003777689198616355, 9.239435621570783541694212063518, 9.490941365860646690337074069155

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