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Properties

Label 2-896-112.27-c1-0-21
Degree $2$
Conductor $896$
Sign $0.710 + 0.703i$
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  + (2.44 − 2.44i)5-s + (1 − 2.44i)7-s + 3i·9-s + (1 − i)11-s + (2.44 + 2.44i)13-s + 4.89i·17-s + (4.89 − 4.89i)19-s − 4·23-s − 6.99i·25-s + (3 − 3i)29-s − 4.89·31-s + (−3.55 − 8.44i)35-s + (−5 − 5i)37-s + 4.89·41-s + (−5 + 5i)43-s + ⋯
L(s)  = 1  + (1.09 − 1.09i)5-s + (0.377 − 0.925i)7-s + i·9-s + (0.301 − 0.301i)11-s + (0.679 + 0.679i)13-s + 1.18i·17-s + (1.12 − 1.12i)19-s − 0.834·23-s − 1.39i·25-s + (0.557 − 0.557i)29-s − 0.879·31-s + (−0.600 − 1.42i)35-s + (−0.821 − 0.821i)37-s + 0.765·41-s + (−0.762 + 0.762i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91707 - 0.788356i\)
\(L(\frac12)\) \(\approx\) \(1.91707 - 0.788356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
good3 \( 1 - 3iT^{2} \)
5 \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + (-4.89 + 4.89i)T - 19iT^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (4.89 + 4.89i)T + 59iT^{2} \)
61 \( 1 + (2.44 + 2.44i)T + 61iT^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−9.973539628988894490678735864558, −9.161436408235113888947957630745, −8.425553498306337441006324865611, −7.58987202082986120495120722851, −6.45712391586217285732445026132, −5.54998655819046276604721595568, −4.75184102580917310702969426332, −3.84658929430287540574155258543, −2.06583562233979210403719824676, −1.18012588529356160786645441024, 1.55171579248739260583583086841, 2.77497670442447878795165404377, 3.58081934858614300328308162650, 5.26939207192183384601391990896, 5.90367560114927729936997235690, 6.64012304767422210633317808484, 7.57500409604876375153472433611, 8.728608395179215842823510742818, 9.512652211741336972134187095380, 10.07822175871216433097689350304

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