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Properties

Label 2-825-5.4-c3-0-77
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  + 0.270i·2-s − 3i·3-s + 7.92·4-s + 0.812·6-s − 33.4i·7-s + 4.31i·8-s − 9·9-s + 11·11-s − 23.7i·12-s + 33.5i·13-s + 9.04·14-s + 62.2·16-s − 71.1i·17-s − 2.43i·18-s + 48.9·19-s + ⋯
L(s)  = 1  + 0.0957i·2-s − 0.577i·3-s + 0.990·4-s + 0.0552·6-s − 1.80i·7-s + 0.190i·8-s − 0.333·9-s + 0.301·11-s − 0.572i·12-s + 0.715i·13-s + 0.172·14-s + 0.972·16-s − 1.01i·17-s − 0.0319i·18-s + 0.591·19-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

\(L(2)\) \(\approx\) \(2.500963358\)
\(L(\frac12)\) \(\approx\) \(2.500963358\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.270iT - 8T^{2} \)
7 \( 1 + 33.4iT - 343T^{2} \)
13 \( 1 - 33.5iT - 2.19e3T^{2} \)
17 \( 1 + 71.1iT - 4.91e3T^{2} \)
19 \( 1 - 48.9T + 6.85e3T^{2} \)
23 \( 1 + 66.7iT - 1.21e4T^{2} \)
29 \( 1 - 66.8T + 2.43e4T^{2} \)
31 \( 1 - 145.T + 2.97e4T^{2} \)
37 \( 1 + 37.8iT - 5.06e4T^{2} \)
41 \( 1 + 344.T + 6.89e4T^{2} \)
43 \( 1 + 34.4iT - 7.95e4T^{2} \)
47 \( 1 + 270. iT - 1.03e5T^{2} \)
53 \( 1 + 666. iT - 1.48e5T^{2} \)
59 \( 1 + 876.T + 2.05e5T^{2} \)
61 \( 1 - 783.T + 2.26e5T^{2} \)
67 \( 1 - 876. iT - 3.00e5T^{2} \)
71 \( 1 + 523.T + 3.57e5T^{2} \)
73 \( 1 + 91.0iT - 3.89e5T^{2} \)
79 \( 1 - 96.9T + 4.93e5T^{2} \)
83 \( 1 - 1.39e3iT - 5.71e5T^{2} \)
89 \( 1 + 508.T + 7.04e5T^{2} \)
97 \( 1 + 644. iT - 9.12e5T^{2} \)
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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

−9.812608386460562288253945553420, −8.462930186764758721018736573540, −7.57458594788074861761966658528, −6.87517853017561575028515410861, −6.58255227380172480750236663735, −5.14031874801430243895612798043, −3.98959417638600560906223347001, −2.90966280773409080645066010986, −1.60861084690180098831285919510, −0.63698376843907699756728789852, 1.53879452056113438302429930086, 2.68739206990591868812017076224, 3.38648203844714888003968245524, 4.91854097806975647386331460173, 5.86064085645638969139956479250, 6.31034811161179417085612563800, 7.69785904008353984148065737335, 8.466321657367688329283688604436, 9.308625663996466343337295848240, 10.15779986054407474782049453553

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