Properties

Label 2-2550-5.4-c1-0-17
Degree $2$
Conductor $2550$
Sign $0.894 - 0.447i$
Analytic cond. $20.3618$
Root an. cond. $4.51241$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s + 4·11-s i·12-s + 2i·13-s + 16-s i·17-s + i·18-s + 4·19-s − 4i·22-s + 4i·23-s − 24-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s + 0.554i·13-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s + 0.917·19-s − 0.852i·22-s + 0.834i·23-s − 0.204·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 17\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(20.3618\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2550} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.682861287\)
\(L(\frac12)\) \(\approx\) \(1.682861287\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 \)
17 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.102686175362718376924115419268, −8.561070558430045159539506325686, −7.42637657357619837608742266578, −6.68157484600785741437958656523, −5.59984040164011358027007334065, −4.93038821072975749988228583327, −3.86206364322777521341628953009, −3.49275402817289348986607960888, −2.20686752998281625449359413655, −1.11452878732348489724099533534, 0.66659293232709081740859685894, 1.86874871860044182921554026911, 3.23934968937428611353574434117, 4.05414912395753202368217812084, 5.15512757928069106842408712785, 5.84623157877719780983481316184, 6.67711219316901086933526183224, 7.17973435817003348097957125423, 8.035609656333998927155692578483, 8.674976872205014579610399513176

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