Properties

Label 2-2550-5.4-c1-0-35
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 + 2i·7-s i·8-s − 9-s i·12-s − 4i·13-s − 2·14-s + 16-s i·17-s i·18-s − 4·19-s − 2·21-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s − 0.917·19-s − 0.436·21-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

\(L(1)\) \(\approx\) \(0.8610221119\)
\(L(\frac12)\) \(\approx\) \(0.8610221119\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.726612759448717761676608627680, −8.245713425672009827702196475572, −7.32700993895320354554923406000, −6.43136247515549689745410535311, −5.65613112164450728550952185048, −5.10887514205157063566702842108, −4.17811415120883478134440091476, −3.23468809516091379113423010245, −2.19495990950516027468695589797, −0.29829319085068413111911254106, 1.22127075589787683742109683507, 2.03246840653963387152554899326, 3.17237105256240963754999452597, 4.10377080365888073746864053291, 4.76779435809111368358280410886, 5.97354125526828108453193730752, 6.66347583389724343324811339243, 7.51253376315177692496404071230, 8.171645687220001427327486960866, 9.110788208929085661582988145167

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