Properties

Degree $2$
Conductor $2550$
Sign $0.894 - 0.447i$
Motivic weight $1$
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 − 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 − 0.204·24-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: $0.894 - 0.447i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(1.256666194\)
\(L(\frac12)\) \(\approx\) \(1.256666194\)
\(L(\frac{3}{2})\) 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 \( 1 - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.087910543834248305861046634847, −8.217657119251712969813260519890, −7.80323405699484606229033304585, −6.48525353000009579257296300101, −5.69750401876247945288874825377, −4.72689083643345171935497120211, −4.30393605622137117591546751804, −2.90052049639930777569910855694, −2.63209985225203183245732193592, −0.940197814882960083574972967081, 0.52462297082912301204589856869, 2.05669617851703378174457535933, 3.00107749095560770941295475832, 4.30483768612326852373799885486, 4.96271772592126370217239940426, 5.99743994823797004689189335285, 6.48349120520564199826544232607, 7.35693212743132773923222440700, 8.013495608938935115657463212527, 8.594596012387730146208750059529

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