Properties

Label 2-825-5.4-c3-0-76
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s − 3i·3-s − 4-s + 9·6-s + 7i·7-s + 21i·8-s − 9·9-s + 11·11-s + 3i·12-s − 16i·13-s − 21·14-s − 71·16-s − 21i·17-s − 27i·18-s − 125·19-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.577i·3-s − 0.125·4-s + 0.612·6-s + 0.377i·7-s + 0.928i·8-s − 0.333·9-s + 0.301·11-s + 0.0721i·12-s − 0.341i·13-s − 0.400·14-s − 1.10·16-s − 0.299i·17-s − 0.353i·18-s − 1.50·19-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: \(1\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT - 8T^{2} \)
7 \( 1 - 7iT - 343T^{2} \)
13 \( 1 + 16iT - 2.19e3T^{2} \)
17 \( 1 + 21iT - 4.91e3T^{2} \)
19 \( 1 + 125T + 6.85e3T^{2} \)
23 \( 1 + 81iT - 1.21e4T^{2} \)
29 \( 1 + 186T + 2.43e4T^{2} \)
31 \( 1 + 58T + 2.97e4T^{2} \)
37 \( 1 - 253iT - 5.06e4T^{2} \)
41 \( 1 - 63T + 6.89e4T^{2} \)
43 \( 1 + 100iT - 7.95e4T^{2} \)
47 \( 1 - 219iT - 1.03e5T^{2} \)
53 \( 1 + 192iT - 1.48e5T^{2} \)
59 \( 1 + 249T + 2.05e5T^{2} \)
61 \( 1 + 64T + 2.26e5T^{2} \)
67 \( 1 + 272iT - 3.00e5T^{2} \)
71 \( 1 + 645T + 3.57e5T^{2} \)
73 \( 1 + 112iT - 3.89e5T^{2} \)
79 \( 1 + 509T + 4.93e5T^{2} \)
83 \( 1 + 1.25e3iT - 5.71e5T^{2} \)
89 \( 1 + 756T + 7.04e5T^{2} \)
97 \( 1 + 839iT - 9.12e5T^{2} \)
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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

−9.155790297706069046263117583171, −8.491962787291807901534593453650, −7.74114880501849830680047366764, −6.89707426528867103012573556693, −6.21771133437610744721269914907, −5.48662081952201599958294920305, −4.37009259755175689019163108613, −2.80895376000091989098846280159, −1.78654587583691056466323434128, 0, 1.50237790942538438773731899179, 2.52696007623732189304681306258, 3.80948539161061902048809567135, 4.18698906397365303528222045677, 5.61931890097627735062426332253, 6.63855070864133296465718794923, 7.55016914545162682355472515475, 8.812013496463059194284410096585, 9.457934176600022399068906861059

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