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Properties

Label 2-825-5.4-c3-0-3
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  − 5.58i·2-s + 3i·3-s − 23.1·4-s + 16.7·6-s + 34.4i·7-s + 84.4i·8-s − 9·9-s + 11·11-s − 69.4i·12-s + 71.2i·13-s + 192.·14-s + 286.·16-s − 22.3i·17-s + 50.2i·18-s − 88.1·19-s + ⋯
L(s)  = 1  − 1.97i·2-s + 0.577i·3-s − 2.89·4-s + 1.13·6-s + 1.85i·7-s + 3.73i·8-s − 0.333·9-s + 0.301·11-s − 1.67i·12-s + 1.52i·13-s + 3.66·14-s + 4.47·16-s − 0.318i·17-s + 0.657i·18-s − 1.06·19-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

\(L(2)\) \(\approx\) \(0.3619019534\)
\(L(\frac12)\) \(\approx\) \(0.3619019534\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.58iT - 8T^{2} \)
7 \( 1 - 34.4iT - 343T^{2} \)
13 \( 1 - 71.2iT - 2.19e3T^{2} \)
17 \( 1 + 22.3iT - 4.91e3T^{2} \)
19 \( 1 + 88.1T + 6.85e3T^{2} \)
23 \( 1 + 21.5iT - 1.21e4T^{2} \)
29 \( 1 + 118.T + 2.43e4T^{2} \)
31 \( 1 + 33.5T + 2.97e4T^{2} \)
37 \( 1 - 364. iT - 5.06e4T^{2} \)
41 \( 1 - 48.9T + 6.89e4T^{2} \)
43 \( 1 + 95.8iT - 7.95e4T^{2} \)
47 \( 1 + 132. iT - 1.03e5T^{2} \)
53 \( 1 + 300. iT - 1.48e5T^{2} \)
59 \( 1 - 654.T + 2.05e5T^{2} \)
61 \( 1 + 772.T + 2.26e5T^{2} \)
67 \( 1 + 112. iT - 3.00e5T^{2} \)
71 \( 1 - 548.T + 3.57e5T^{2} \)
73 \( 1 + 559. iT - 3.89e5T^{2} \)
79 \( 1 + 48.5T + 4.93e5T^{2} \)
83 \( 1 + 447. iT - 5.71e5T^{2} \)
89 \( 1 - 552.T + 7.04e5T^{2} \)
97 \( 1 + 413. iT - 9.12e5T^{2} \)
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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

−10.12754033433142513255943127298, −9.316411271947829024004705684228, −8.959787288344253050979059713922, −8.297532959872949880652686052547, −6.27529502697213184042764216015, −5.19751772871372978478624460495, −4.48165264849601429695508340762, −3.48190848375504143075822131466, −2.42058106148993245896441180675, −1.78347524130547476582457536545, 0.11877992852208674548357863349, 0.987852659991446679850182491038, 3.60938431948111450002299416115, 4.31638553130446890505176480450, 5.46202333306086147979987799525, 6.26898420015121193241630434346, 7.09640003751128333858771622299, 7.64840513927168871534239567036, 8.202374915258579245113100599422, 9.225121730921516934774909697626

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