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Properties

Label 2-825-5.4-c3-0-51
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  + 0.607i·2-s + 3i·3-s + 7.63·4-s − 1.82·6-s − 8.95i·7-s + 9.49i·8-s − 9·9-s − 11·11-s + 22.8i·12-s − 0.460i·13-s + 5.43·14-s + 55.2·16-s − 128. i·17-s − 5.46i·18-s + 0.0245·19-s + ⋯
L(s)  = 1  + 0.214i·2-s + 0.577i·3-s + 0.953·4-s − 0.123·6-s − 0.483i·7-s + 0.419i·8-s − 0.333·9-s − 0.301·11-s + 0.550i·12-s − 0.00982i·13-s + 0.103·14-s + 0.863·16-s − 1.83i·17-s − 0.0715i·18-s + 0.000296·19-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

\(L(2)\) \(\approx\) \(2.690757164\)
\(L(\frac12)\) \(\approx\) \(2.690757164\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.607iT - 8T^{2} \)
7 \( 1 + 8.95iT - 343T^{2} \)
13 \( 1 + 0.460iT - 2.19e3T^{2} \)
17 \( 1 + 128. iT - 4.91e3T^{2} \)
19 \( 1 - 0.0245T + 6.85e3T^{2} \)
23 \( 1 - 171. iT - 1.21e4T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 - 195.T + 2.97e4T^{2} \)
37 \( 1 + 338. iT - 5.06e4T^{2} \)
41 \( 1 - 136.T + 6.89e4T^{2} \)
43 \( 1 - 336. iT - 7.95e4T^{2} \)
47 \( 1 - 540. iT - 1.03e5T^{2} \)
53 \( 1 + 622. iT - 1.48e5T^{2} \)
59 \( 1 - 9.86T + 2.05e5T^{2} \)
61 \( 1 - 902.T + 2.26e5T^{2} \)
67 \( 1 + 146. iT - 3.00e5T^{2} \)
71 \( 1 + 893.T + 3.57e5T^{2} \)
73 \( 1 + 1.14e3iT - 3.89e5T^{2} \)
79 \( 1 - 459.T + 4.93e5T^{2} \)
83 \( 1 + 125. iT - 5.71e5T^{2} \)
89 \( 1 + 150.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3iT - 9.12e5T^{2} \)
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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

−9.939014895109082640508617283637, −9.212462421796047129888106206627, −7.967983073150908351558447601554, −7.37070891614061318582884119103, −6.47376592657707309448948610961, −5.47350185148753963771224650982, −4.61137678799754736338023921195, −3.30874925577233459273452098192, −2.46866467723217251231414210526, −0.879673368876732858379149746925, 0.974564295069225120970765861930, 2.15300725770004518742271538447, 2.90020609253332099696020127843, 4.23565104299634918964982004169, 5.63661203326523670545761900965, 6.39362515101657622970359050731, 7.00922298970735205691901182528, 8.293833088752869447769044072936, 8.509867396350671962701709657919, 10.23873283365539385627584660444

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