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Properties

Label 2-896-16.13-c1-0-8
Degree $2$
Conductor $896$
Sign $-0.273 + 0.961i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.05 − 2.05i)3-s + (−2.72 + 2.72i)5-s + i·7-s + 5.44i·9-s + (0.919 − 0.919i)11-s + (1.12 + 1.12i)13-s + 11.2·15-s − 1.50·17-s + (1.46 + 1.46i)19-s + (2.05 − 2.05i)21-s − 4.77i·23-s − 9.88i·25-s + (5.02 − 5.02i)27-s + (−4.10 − 4.10i)29-s − 4.10·31-s + ⋯
L(s)  = 1  + (−1.18 − 1.18i)3-s + (−1.21 + 1.21i)5-s + 0.377i·7-s + 1.81i·9-s + (0.277 − 0.277i)11-s + (0.312 + 0.312i)13-s + 2.89·15-s − 0.365·17-s + (0.335 + 0.335i)19-s + (0.448 − 0.448i)21-s − 0.994i·23-s − 1.97i·25-s + (0.967 − 0.967i)27-s + (−0.761 − 0.761i)29-s − 0.738·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.273 + 0.961i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273309 - 0.361735i\)
\(L(\frac12)\) \(\approx\) \(0.273309 - 0.361735i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + (2.05 + 2.05i)T + 3iT^{2} \)
5 \( 1 + (2.72 - 2.72i)T - 5iT^{2} \)
11 \( 1 + (-0.919 + 0.919i)T - 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + (-1.46 - 1.46i)T + 19iT^{2} \)
23 \( 1 + 4.77iT - 23T^{2} \)
29 \( 1 + (4.10 + 4.10i)T + 29iT^{2} \)
31 \( 1 + 4.10T + 31T^{2} \)
37 \( 1 + (-1.65 + 1.65i)T - 37iT^{2} \)
41 \( 1 - 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + 3.59T + 47T^{2} \)
53 \( 1 + (-0.675 + 0.675i)T - 53iT^{2} \)
59 \( 1 + (-1.13 + 1.13i)T - 59iT^{2} \)
61 \( 1 + (-3.21 - 3.21i)T + 61iT^{2} \)
67 \( 1 + (1.52 + 1.52i)T + 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 + (7.16 + 7.16i)T + 83iT^{2} \)
89 \( 1 + 8.45iT - 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24675223656382216586752818047, −8.812755456926292808743464018755, −7.76260758572728998235950706223, −7.28569435032062072953503688039, −6.40933435484268658062995562583, −5.93516050126705452853076200558, −4.54862103925139385291771671783, −3.37562306467592144112169168015, −2.05512416542611435860937389947, −0.32799072724882432446680208482, 0.996450181514775343115952667795, 3.64965107323576048912785380606, 4.14499590285030342545938325996, 5.04284438819458301740564272322, 5.60333171074241454105627312744, 6.94612748476023534091213842602, 7.85982054957725624622403122828, 8.946414835885707570189894269063, 9.485755787718656511699799588441, 10.50155955767418270065122055511

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