↑

Properties

Label 2-2550-5.4-c1-0-40
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 $1$

Origins

Related objects

Downloads

Learn more

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 + 6i·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 + 1.66i·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: \(1\)
Selberg data: \((2,\ 2550,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 6iT - 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.825097305600924684882846311536, −7.83130204114576913522229840721, −6.84691333396427191508576090187, −6.19858416419371220177785976867, −4.88585937677311203950539035886, −4.37309391978305386506785776826, −3.70998212691240318264645158834, −2.59039265610919749591723163296, −1.56278854575909790660249147422, 0, 1.53706271652864137809279495870, 2.78147945032466614126753530538, 3.65469738875855982980400887343, 4.96176379420631685054632513019, 5.61740178378633776751199086807, 6.17815197663111214311890216504, 7.11323696493776343337880421612, 7.78367525758314996864358333183, 8.495745477700653924604162293949

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