↑

Properties

Label 2-1456-7.2-c1-0-3
Degree $2$
Conductor $1456$
Sign $-0.930 - 0.366i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.129 − 0.224i)3-s + (−1.96 + 3.40i)5-s + (−1.12 + 2.39i)7-s + (1.46 − 2.53i)9-s + (2.25 + 3.90i)11-s + 13-s + 1.02·15-s + (1.14 + 1.97i)17-s + (−0.893 + 1.54i)19-s + (0.684 − 0.0584i)21-s + (0.870 − 1.50i)23-s + (−5.23 − 9.06i)25-s − 1.54·27-s + 1.65·29-s + (2.80 + 4.85i)31-s + ⋯
L(s)  = 1  + (−0.0749 − 0.129i)3-s + (−0.879 + 1.52i)5-s + (−0.424 + 0.905i)7-s + (0.488 − 0.846i)9-s + (0.679 + 1.17i)11-s + 0.277·13-s + 0.263·15-s + (0.276 + 0.479i)17-s + (−0.205 + 0.355i)19-s + (0.149 − 0.0127i)21-s + (0.181 − 0.314i)23-s + (−1.04 − 1.81i)25-s − 0.296·27-s + 0.306·29-s + (0.503 + 0.871i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.930 - 0.366i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.930 - 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9599820127\)
\(L(\frac12)\) \(\approx\) \(0.9599820127\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.12 - 2.39i)T \)
13 \( 1 - T \)
good3 \( 1 + (0.129 + 0.224i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.96 - 3.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.14 - 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.893 - 1.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.870 + 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.65T + 29T^{2} \)
31 \( 1 + (-2.80 - 4.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 - 6.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 + 6.81T + 43T^{2} \)
47 \( 1 + (-1.77 + 3.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.64 + 2.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.25 - 3.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.77 - 6.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.33 + 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.54T + 71T^{2} \)
73 \( 1 + (0.540 + 0.935i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.395 + 0.685i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 + (-5.63 + 9.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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

−10.14293058521220124643748226187, −9.062509498106337990160094301516, −8.235967803900830104690184719561, −7.19824992451430403101502515689, −6.65480611934235560430615160976, −6.18522476441277465418683691606, −4.70721232335956604274134009425, −3.65520862745183164804300437623, −3.08355414119138463627764714100, −1.76648439390584012026586909428, 0.41543807379306541289208704521, 1.41245442244693739202744027189, 3.33043529695255119436124108628, 4.13175613148574788192560618717, 4.78844613051782581488216140571, 5.69929841960543209293903714697, 6.85761988794536376426138246034, 7.70625619036236973447979958040, 8.362706218351914073775786835287, 9.058894476821846308409096074056

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