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Properties

Label 2-825-5.4-c3-0-52
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  + 4.38i·2-s − 3i·3-s − 11.2·4-s + 13.1·6-s + 11.7i·7-s − 14.2i·8-s − 9·9-s + 11·11-s + 33.7i·12-s − 72.8i·13-s − 51.4·14-s − 27.3·16-s + 9.89i·17-s − 39.4i·18-s − 0.0238·19-s + ⋯
L(s)  = 1  + 1.55i·2-s − 0.577i·3-s − 1.40·4-s + 0.895·6-s + 0.633i·7-s − 0.631i·8-s − 0.333·9-s + 0.301·11-s + 0.812i·12-s − 1.55i·13-s − 0.982·14-s − 0.426·16-s + 0.141i·17-s − 0.517i·18-s − 0.000288·19-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

\(L(2)\) \(\approx\) \(1.470245304\)
\(L(\frac12)\) \(\approx\) \(1.470245304\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.38iT - 8T^{2} \)
7 \( 1 - 11.7iT - 343T^{2} \)
13 \( 1 + 72.8iT - 2.19e3T^{2} \)
17 \( 1 - 9.89iT - 4.91e3T^{2} \)
19 \( 1 + 0.0238T + 6.85e3T^{2} \)
23 \( 1 - 73.0iT - 1.21e4T^{2} \)
29 \( 1 + 202.T + 2.43e4T^{2} \)
31 \( 1 - 181.T + 2.97e4T^{2} \)
37 \( 1 + 299. iT - 5.06e4T^{2} \)
41 \( 1 - 88.5T + 6.89e4T^{2} \)
43 \( 1 + 146. iT - 7.95e4T^{2} \)
47 \( 1 + 185. iT - 1.03e5T^{2} \)
53 \( 1 + 347. iT - 1.48e5T^{2} \)
59 \( 1 + 691.T + 2.05e5T^{2} \)
61 \( 1 - 491.T + 2.26e5T^{2} \)
67 \( 1 + 715. iT - 3.00e5T^{2} \)
71 \( 1 - 541.T + 3.57e5T^{2} \)
73 \( 1 + 159. iT - 3.89e5T^{2} \)
79 \( 1 - 212.T + 4.93e5T^{2} \)
83 \( 1 - 413. iT - 5.71e5T^{2} \)
89 \( 1 - 1.09e3T + 7.04e5T^{2} \)
97 \( 1 + 567. iT - 9.12e5T^{2} \)
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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

−9.476800366853235486167951393374, −8.732027776229399882564911168074, −7.910504469344984720289546022607, −7.42177892801253557396638038632, −6.38474898242672282771120981820, −5.70191986036564113213159354284, −5.09322332178510061089679187157, −3.59569481581136268637450347590, −2.20446374718545994646744891923, −0.46888439050595172039664651495, 0.984575973238767577934621461867, 2.12017290938942708989069213662, 3.24578303798602219796874319867, 4.21859974898862005948027952461, 4.64799749478372910708621657688, 6.23387546928713602285425681578, 7.20037113926887835128432236946, 8.538202627054264409292146970457, 9.342139038748520502213393055616, 9.878796884349014903338724412477

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