Properties

Label 2-825-5.4-c3-0-56
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  + 5.17i·2-s − 3i·3-s − 18.7·4-s + 15.5·6-s − 11.1i·7-s − 55.5i·8-s − 9·9-s − 11·11-s + 56.2i·12-s + 89.5i·13-s + 57.7·14-s + 137.·16-s + 58.3i·17-s − 46.5i·18-s − 24.5·19-s + ⋯
L(s)  = 1  + 1.82i·2-s − 0.577i·3-s − 2.34·4-s + 1.05·6-s − 0.603i·7-s − 2.45i·8-s − 0.333·9-s − 0.301·11-s + 1.35i·12-s + 1.91i·13-s + 1.10·14-s + 2.14·16-s + 0.832i·17-s − 0.609i·18-s − 0.296·19-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

\(L(2)\) \(\approx\) \(0.4022225758\)
\(L(\frac12)\) \(\approx\) \(0.4022225758\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.17iT - 8T^{2} \)
7 \( 1 + 11.1iT - 343T^{2} \)
13 \( 1 - 89.5iT - 2.19e3T^{2} \)
17 \( 1 - 58.3iT - 4.91e3T^{2} \)
19 \( 1 + 24.5T + 6.85e3T^{2} \)
23 \( 1 - 111. iT - 1.21e4T^{2} \)
29 \( 1 + 109.T + 2.43e4T^{2} \)
31 \( 1 - 119.T + 2.97e4T^{2} \)
37 \( 1 + 356. iT - 5.06e4T^{2} \)
41 \( 1 - 268.T + 6.89e4T^{2} \)
43 \( 1 + 263. iT - 7.95e4T^{2} \)
47 \( 1 + 206. iT - 1.03e5T^{2} \)
53 \( 1 + 223. iT - 1.48e5T^{2} \)
59 \( 1 + 475.T + 2.05e5T^{2} \)
61 \( 1 + 513.T + 2.26e5T^{2} \)
67 \( 1 + 264. iT - 3.00e5T^{2} \)
71 \( 1 + 1.11e3T + 3.57e5T^{2} \)
73 \( 1 + 893. iT - 3.89e5T^{2} \)
79 \( 1 + 1.30e3T + 4.93e5T^{2} \)
83 \( 1 + 1.04e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 85.8iT - 9.12e5T^{2} \)
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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

−9.207986736901024591548404530573, −8.803032515594498542745755273399, −7.59958078173068009386751355596, −7.31891582463420125853575505096, −6.38257382473602633589350107138, −5.76348081453567480240514564647, −4.56837133144743511436466747660, −3.82984163532590855436232820584, −1.76399549272046458634304099687, −0.12581489692424302992692394892, 1.03424058800453668267591042487, 2.73983976282850732046405370575, 2.90974158738069030016261847770, 4.28804200315308402462859274428, 5.06040216657302902739835210028, 5.95743955640675832096976619853, 7.80828370363865611230752392936, 8.596363582038144719769986251340, 9.380682976933471069859707254448, 10.17416082691230086476709381687

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