↑

Properties

Label 2-896-112.109-c1-0-27
Degree $2$
Conductor $896$
Sign $-0.997 + 0.0679i$
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  + (1.60 − 0.430i)3-s + (−1.94 − 0.522i)5-s + (−1.89 + 1.85i)7-s + (−0.196 + 0.113i)9-s + (−1.63 − 6.09i)11-s + (−1.13 − 1.13i)13-s − 3.35·15-s + (−0.960 + 1.66i)17-s + (−1.63 + 6.09i)19-s + (−2.24 + 3.79i)21-s + (−0.924 + 0.533i)23-s + (−0.804 − 0.464i)25-s + (−3.79 + 3.79i)27-s + (−5.08 − 5.08i)29-s + (−0.198 + 0.343i)31-s + ⋯
L(s)  = 1  + (0.928 − 0.248i)3-s + (−0.871 − 0.233i)5-s + (−0.714 + 0.699i)7-s + (−0.0655 + 0.0378i)9-s + (−0.492 − 1.83i)11-s + (−0.314 − 0.314i)13-s − 0.867·15-s + (−0.233 + 0.403i)17-s + (−0.374 + 1.39i)19-s + (−0.489 + 0.827i)21-s + (−0.192 + 0.111i)23-s + (−0.160 − 0.0928i)25-s + (−0.731 + 0.731i)27-s + (−0.945 − 0.945i)29-s + (−0.0356 + 0.0616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.997 + 0.0679i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.997 + 0.0679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00881009 - 0.258930i\)
\(L(\frac12)\) \(\approx\) \(0.00881009 - 0.258930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 - 1.85i)T \)
good3 \( 1 + (-1.60 + 0.430i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.94 + 0.522i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.63 + 6.09i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.13 + 1.13i)T + 13iT^{2} \)
17 \( 1 + (0.960 - 1.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 - 6.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.924 - 0.533i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.08 + 5.08i)T + 29iT^{2} \)
31 \( 1 + (0.198 - 0.343i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.327 + 0.0877i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.26iT - 41T^{2} \)
43 \( 1 + (1.75 - 1.75i)T - 43iT^{2} \)
47 \( 1 + (1.08 + 1.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.111 + 0.415i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.32 - 8.67i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.72 + 10.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.87 - 2.10i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.78iT - 71T^{2} \)
73 \( 1 + (-2.05 - 1.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.17 + 1.17i)T + 83iT^{2} \)
89 \( 1 + (-11.0 + 6.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.21T + 97T^{2} \)
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

−9.509477720502924653576464562338, −8.475646659071752026313833918224, −8.318121640149260802322180141616, −7.49192380837928435445167456780, −6.07617657720905269662290495779, −5.53931381490459655184659934194, −3.84901713273039668979755148896, −3.27567119775285607491852284730, −2.20884541567419315849320759584, −0.10032860353281945736331797799, 2.23992852020195747228258699310, 3.22319418840159930241956503605, 4.14838242978459654493904694954, 4.89787493555755274184067254513, 6.59813913480012196996836249783, 7.30326940367950854671090161210, 7.81912630719186594334257650021, 9.047679953444697868585793542450, 9.542212277104231509527766329003, 10.35912599912174529059960816941

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