↑

Properties

Label 2-2800-7.6-c0-0-0
Degree $2$
Conductor $2800$
Sign $-i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + i·7-s + 11-s − i·13-s + i·17-s − 21-s + i·27-s + 29-s + i·33-s + 39-s − i·47-s − 49-s − 51-s − 2·71-s + 2i·73-s + ⋯
L(s)  = 1  + i·3-s + i·7-s + 11-s − i·13-s + i·17-s − 21-s + i·27-s + 29-s + i·33-s + 39-s − i·47-s − 49-s − 51-s − 2·71-s + 2i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344673307\)
\(L(\frac12)\) \(\approx\) \(1.344673307\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 - iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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

−9.026368704389947980846614542107, −8.719092726439537369148181610955, −7.83577241949867935734164701876, −6.76720951016542946468186860957, −5.97417910944711007300441616314, −5.28996130437705788064625617305, −4.42400484698112915864487358015, −3.66365492791129336934090479355, −2.80607882101970589734710714815, −1.51528762134580811113057519455, 0.980421522165142962586538581285, 1.80012982497458135385205880478, 3.02910236435437842058177517255, 4.17542394167422017363378338004, 4.66904713183137169292203587462, 6.05700980466557834257382444348, 6.73690912744666326733968084455, 7.15288482848796159652818718685, 7.82013759181595875966845514641, 8.763040401577450517067061640054

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