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Properties

Label 2-896-56.3-c1-0-8
Degree $2$
Conductor $896$
Sign $0.293 - 0.955i$
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  + (2.63 + 1.52i)3-s + (−0.866 − 1.5i)5-s + (−2.14 + 1.54i)7-s + (3.12 + 5.40i)9-s + (−1.09 + 1.88i)11-s + 5.91·13-s − 5.26i·15-s + (3.62 + 2.09i)17-s + (−3.27 + 1.88i)19-s + (−8.00 + 0.792i)21-s + (1.88 − 1.09i)23-s + (1 − 1.73i)25-s + 9.85i·27-s + 7.41i·29-s + (−4.03 + 6.99i)31-s + ⋯
L(s)  = 1  + (1.52 + 0.877i)3-s + (−0.387 − 0.670i)5-s + (−0.812 + 0.582i)7-s + (1.04 + 1.80i)9-s + (−0.328 + 0.569i)11-s + 1.64·13-s − 1.35i·15-s + (0.878 + 0.507i)17-s + (−0.750 + 0.433i)19-s + (−1.74 + 0.173i)21-s + (0.393 − 0.227i)23-s + (0.200 − 0.346i)25-s + 1.89i·27-s + 1.37i·29-s + (−0.725 + 1.25i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87769 + 1.38737i\)
\(L(\frac12)\) \(\approx\) \(1.87769 + 1.38737i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 1.54i)T \)
good3 \( 1 + (-2.63 - 1.52i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.09 - 1.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + (-3.62 - 2.09i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 - 1.88i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.88 + 1.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.41iT - 29T^{2} \)
31 \( 1 + (4.03 - 6.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.07 - 0.621i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.34iT - 41T^{2} \)
43 \( 1 - 6.16T + 43T^{2} \)
47 \( 1 + (4.56 + 7.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.37 + 0.792i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.63 - 1.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.81 - 8.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.72iT - 71T^{2} \)
73 \( 1 + (11.7 + 6.77i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.23 + 4.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (-9.98 + 5.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.91iT - 97T^{2} \)
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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

−10.32900725283481105619045820659, −9.052160512822785685523533532916, −8.803064000341656528216639974506, −8.197581612518198583898224176561, −7.08830237581775975217476738175, −5.80323502722252026323192924686, −4.72196487742199301529781488365, −3.70259131916046502797307873215, −3.18708979719854017759382953933, −1.78529340308732498458447659117, 1.01889314604101194550910439550, 2.60487697524821674910955332050, 3.35226668955189181854652578014, 3.98599739349653472245212866457, 6.01320186087410664114851531018, 6.69252889456456377762072136653, 7.61185056194423124298983586557, 8.050105507161118072172381489794, 9.039100410891889533117497424580, 9.684517832252652822504240467864

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