Properties

Label 2-1400-5.4-c1-0-9
Degree $2$
Conductor $1400$
Sign $0.447 - 0.894i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s i·7-s + 2·9-s − 5·11-s i·13-s + 3i·17-s + 6·19-s + 21-s + 6i·23-s + 5i·27-s + 9·29-s − 5i·33-s + 6i·37-s + 39-s + 8·41-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.377i·7-s + 0.666·9-s − 1.50·11-s − 0.277i·13-s + 0.727i·17-s + 1.37·19-s + 0.218·21-s + 1.25i·23-s + 0.962i·27-s + 1.67·29-s − 0.870i·33-s + 0.986i·37-s + 0.160·39-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.603190838\)
\(L(\frac12)\) \(\approx\) \(1.603190838\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - iT - 3T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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

−9.830355631542343965210369418935, −9.074472203004943738961237653960, −7.77483863312673220349810341921, −7.64451692575950282623666912201, −6.38508060345397383102242753684, −5.33803774843133144785432928121, −4.73191917989119424612534923405, −3.66694085477657493600548533871, −2.77379127881215527383193628069, −1.21516411902723416399131658557, 0.75426739068348235660808495487, 2.24087465779208734826785086378, 3.01517848820079930900166197188, 4.50306578537665513108174397465, 5.19725513004886246870487322093, 6.21194920886477618236243061206, 7.09863865729534740582752256860, 7.74152208327946008940158465062, 8.454319508032720097249076904898, 9.511146756694063072849780364616

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