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Properties

Label 2-825-5.4-c3-0-88
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  − 4.08i·2-s − 3i·3-s − 8.69·4-s − 12.2·6-s − 7.95i·7-s + 2.82i·8-s − 9·9-s − 11·11-s + 26.0i·12-s − 47.2i·13-s − 32.5·14-s − 57.9·16-s − 82.2i·17-s + 36.7i·18-s + 18.3·19-s + ⋯
L(s)  = 1  − 1.44i·2-s − 0.577i·3-s − 1.08·4-s − 0.833·6-s − 0.429i·7-s + 0.124i·8-s − 0.333·9-s − 0.301·11-s + 0.627i·12-s − 1.00i·13-s − 0.620·14-s − 0.906·16-s − 1.17i·17-s + 0.481i·18-s + 0.221·19-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

\(L(2)\) \(\approx\) \(1.066353799\)
\(L(\frac12)\) \(\approx\) \(1.066353799\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.08iT - 8T^{2} \)
7 \( 1 + 7.95iT - 343T^{2} \)
13 \( 1 + 47.2iT - 2.19e3T^{2} \)
17 \( 1 + 82.2iT - 4.91e3T^{2} \)
19 \( 1 - 18.3T + 6.85e3T^{2} \)
23 \( 1 + 128. iT - 1.21e4T^{2} \)
29 \( 1 - 84.1T + 2.43e4T^{2} \)
31 \( 1 - 51.0T + 2.97e4T^{2} \)
37 \( 1 + 80.6iT - 5.06e4T^{2} \)
41 \( 1 + 482.T + 6.89e4T^{2} \)
43 \( 1 + 223. iT - 7.95e4T^{2} \)
47 \( 1 - 305. iT - 1.03e5T^{2} \)
53 \( 1 - 606. iT - 1.48e5T^{2} \)
59 \( 1 + 122.T + 2.05e5T^{2} \)
61 \( 1 - 13.8T + 2.26e5T^{2} \)
67 \( 1 - 795. iT - 3.00e5T^{2} \)
71 \( 1 + 99.8T + 3.57e5T^{2} \)
73 \( 1 - 12.2iT - 3.89e5T^{2} \)
79 \( 1 + 180.T + 4.93e5T^{2} \)
83 \( 1 + 144. iT - 5.71e5T^{2} \)
89 \( 1 - 832.T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3iT - 9.12e5T^{2} \)
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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

−9.366339661208117330045412907896, −8.456648247781272743506272424340, −7.46556257054997931512662904755, −6.63870747732727047743075815243, −5.34973682700961865348812536674, −4.33241184028112508050075797584, −3.11000320656086585034854002908, −2.48217958926829577365756321847, −1.12729518335862016871286943114, −0.30335024311426971558055899582, 1.93151843653830611966990037260, 3.48329706977220333781388308256, 4.62261287088498395190190572090, 5.38431928275796676234040304775, 6.23071347560692269385415070656, 6.96720907382101910925257772192, 8.038742642899700955411365896379, 8.613090088983408232320220011561, 9.430327983708811069234577559988, 10.28860124114147530030045913280

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