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Properties

Label 2-825-5.4-c3-0-69
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  + 3.59i·2-s + 3i·3-s − 4.89·4-s − 10.7·6-s − 16.1i·7-s + 11.1i·8-s − 9·9-s + 11·11-s − 14.6i·12-s + 54.1i·13-s + 57.9·14-s − 79.2·16-s − 107. i·17-s − 32.3i·18-s − 48.7·19-s + ⋯
L(s)  = 1  + 1.26i·2-s + 0.577i·3-s − 0.611·4-s − 0.732·6-s − 0.871i·7-s + 0.493i·8-s − 0.333·9-s + 0.301·11-s − 0.353i·12-s + 1.15i·13-s + 1.10·14-s − 1.23·16-s − 1.52i·17-s − 0.423i·18-s − 0.588·19-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

\(L(2)\) \(\approx\) \(0.5719441944\)
\(L(\frac12)\) \(\approx\) \(0.5719441944\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.59iT - 8T^{2} \)
7 \( 1 + 16.1iT - 343T^{2} \)
13 \( 1 - 54.1iT - 2.19e3T^{2} \)
17 \( 1 + 107. iT - 4.91e3T^{2} \)
19 \( 1 + 48.7T + 6.85e3T^{2} \)
23 \( 1 + 11.9iT - 1.21e4T^{2} \)
29 \( 1 + 239.T + 2.43e4T^{2} \)
31 \( 1 + 82.0T + 2.97e4T^{2} \)
37 \( 1 + 21.7iT - 5.06e4T^{2} \)
41 \( 1 + 124.T + 6.89e4T^{2} \)
43 \( 1 + 224. iT - 7.95e4T^{2} \)
47 \( 1 + 186. iT - 1.03e5T^{2} \)
53 \( 1 + 233. iT - 1.48e5T^{2} \)
59 \( 1 + 232.T + 2.05e5T^{2} \)
61 \( 1 - 163.T + 2.26e5T^{2} \)
67 \( 1 + 876. iT - 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3iT - 3.89e5T^{2} \)
79 \( 1 - 588.T + 4.93e5T^{2} \)
83 \( 1 - 1.16e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.04e3T + 7.04e5T^{2} \)
97 \( 1 - 1.54e3iT - 9.12e5T^{2} \)
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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

−9.390462989063643004273335217965, −8.997267998056960619072919001887, −7.82579541193419336610784081188, −7.10059928253472867089714288081, −6.49870885328701099282406900903, −5.37169642272928413051914471634, −4.58685377708898067915749473034, −3.66750897630696223020547677680, −2.07141164971225816415223993932, −0.14572755111208684153193827265, 1.30572348185022477284402139914, 2.17489588850634757249021188060, 3.14235106942633049239877996582, 4.09913153835463025329366251626, 5.56987505203794453990543621571, 6.26520819058045727158259345725, 7.43406930515446371120130445826, 8.415960758764556925244101400029, 9.144086565267330360712416621209, 10.13605499264659932894403024429

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