Properties

Label 2-896-56.19-c1-0-25
Degree $2$
Conductor $896$
Sign $0.367 + 0.929i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (1.20 − 0.696i)3-s + (2.07 − 3.59i)5-s + (0.632 + 2.56i)7-s + (−0.530 + 0.919i)9-s + (−2.36 − 4.09i)11-s + 3.68·13-s − 5.78i·15-s + (2.28 − 1.31i)17-s + (−0.698 − 0.403i)19-s + (2.55 + 2.65i)21-s + (4.73 + 2.73i)23-s + (−6.13 − 10.6i)25-s + 5.65i·27-s − 3.88i·29-s + (0.515 + 0.893i)31-s + ⋯
L(s)  = 1  + (0.696 − 0.401i)3-s + (0.929 − 1.60i)5-s + (0.239 + 0.970i)7-s + (−0.176 + 0.306i)9-s + (−0.712 − 1.23i)11-s + 1.02·13-s − 1.49i·15-s + (0.553 − 0.319i)17-s + (−0.160 − 0.0924i)19-s + (0.556 + 0.579i)21-s + (0.988 + 0.570i)23-s + (−1.22 − 2.12i)25-s + 1.08i·27-s − 0.721i·29-s + (0.0926 + 0.160i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92796 - 1.31084i\)
\(L(\frac12)\) \(\approx\) \(1.92796 - 1.31084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.632 - 2.56i)T \)
good3 \( 1 + (-1.20 + 0.696i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.07 + 3.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (-2.28 + 1.31i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.698 + 0.403i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.73 - 2.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.88iT - 29T^{2} \)
31 \( 1 + (-0.515 - 0.893i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.03 - 0.596i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.47iT - 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (3.98 - 6.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.32 - 1.34i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.38 + 1.37i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.819 + 1.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.45 - 4.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.2 - 7.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.37iT - 83T^{2} \)
89 \( 1 + (-15.8 - 9.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.83iT - 97T^{2} \)
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

−9.593266941650635472425756652453, −8.902540909924969107530193943987, −8.416778009064248212399401494854, −7.901037631973303848813514006956, −6.23066121183501974330411716109, −5.46670985237781621918222239760, −4.98567980570390478045633022556, −3.28892118725845759577947776111, −2.21514353279641639423152154803, −1.13946585735457863594537201572, 1.79146997580787478187300085555, 2.98120955833277054999024330495, 3.62673213754514327991561611082, 4.89154467508405954076087948796, 6.19159007649410431211778852863, 6.85102878119453812567806406434, 7.64734173159996874275297237993, 8.645573376492164096798802836651, 9.765986914580209546602346592669, 10.19454851379759354946009276074

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