↑

Properties

Label 2-2800-35.34-c0-0-0
Degree $2$
Conductor $2800$
Sign $0.447 - 0.894i$
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·7-s − 9-s + 11-s + i·23-s + 29-s + i·37-s + i·43-s − 49-s + 2i·53-s − i·63-s − i·67-s + 71-s + i·77-s − 79-s + 81-s + ⋯
L(s)  = 1  + i·7-s − 9-s + 11-s + i·23-s + 29-s + i·37-s + i·43-s − 49-s + 2i·53-s − i·63-s − i·67-s + 71-s + i·77-s − 79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.124035799\)
\(L(\frac12)\) \(\approx\) \(1.124035799\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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

−9.135171382778324257841071363221, −8.435144917375820674621288927322, −7.77869830947886379175377196948, −6.63805530292510685727169734775, −6.08377653563261130061383429280, −5.35715730028588946782641814349, −4.47416567204505404622181264149, −3.33801673320425637122490388489, −2.64203220024062363062844504018, −1.43015267771404055818844501684, 0.76433077461888908712731207369, 2.15721943027365585074170429905, 3.29801930707782801499375189148, 4.04458311518782850130674061207, 4.88573245467035886049476879301, 5.87398772597240078277542629910, 6.64852555617547412689868432602, 7.20956730405814256119774091571, 8.278528101102618437396936855942, 8.693740333959821457089854034731

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