↑

Properties

Label 2-2850-5.4-c1-0-55
Degree $2$
Conductor $2850$
Sign $0.447 - 0.894i$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
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 − 4i·7-s + i·8-s − 9-s − 4·11-s + i·12-s + 6i·13-s − 4·14-s + 16-s − 6i·17-s + i·18-s + 19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 1.66i·13-s − 1.06·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(22.7573\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2850} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 0.447 - 0.894i)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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

−7.998101945442112101097968290688, −7.39169658548347539878920690744, −6.92792546672046921019481485190, −5.82864038938981792931961554854, −4.74056725615326966757794434412, −4.23028916231228240379328945862, −3.12979719336492174869604714731, −2.22918500235320419014572217662, −1.14966679848440505616067908841, 0, 2.05515554963146575592832786288, 3.07851030095443522553505219361, 3.88073497093061971168600235641, 5.19697352013823300227655925677, 5.68057094767375485779222580700, 5.83154338404942666368037420418, 7.38711328891432270106239604100, 7.88700471717932672810990170542, 8.703273392880148594582583378676

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