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Properties

Label 1-896-896.173-r1-0-0
Degree $1$
Conductor $896$
Sign $0.215 - 0.976i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.997 + 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (0.896 − 0.442i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.321 + 0.946i)19-s + (−0.130 − 0.991i)23-s + (0.130 − 0.991i)25-s + (−0.980 + 0.195i)27-s + (0.831 − 0.555i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.659 + 0.751i)37-s + ⋯
L(s)  = 1  + (−0.997 + 0.0654i)3-s + (−0.751 + 0.659i)5-s + (0.991 − 0.130i)9-s + (0.896 − 0.442i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.965 − 0.258i)17-s + (0.321 + 0.946i)19-s + (−0.130 − 0.991i)23-s + (0.130 − 0.991i)25-s + (−0.980 + 0.195i)27-s + (0.831 − 0.555i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.659 + 0.751i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.215 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8157100021 - 0.6555519533i\)
\(L(\frac12)\) \(\approx\) \(0.8157100021 - 0.6555519533i\)
\(L(1)\) \(\approx\) \(0.7587425983 - 0.03506926379i\)
\(L(1)\) \(\approx\) \(0.7587425983 - 0.03506926379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.997 + 0.0654i)T \)
5 \( 1 + (-0.751 + 0.659i)T \)
11 \( 1 + (0.896 - 0.442i)T \)
13 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (0.321 + 0.946i)T \)
23 \( 1 + (-0.130 - 0.991i)T \)
29 \( 1 + (0.831 - 0.555i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.659 + 0.751i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.555 + 0.831i)T \)
47 \( 1 + (0.258 - 0.965i)T \)
53 \( 1 + (-0.896 + 0.442i)T \)
59 \( 1 + (-0.946 - 0.321i)T \)
61 \( 1 + (-0.442 + 0.896i)T \)
67 \( 1 + (-0.997 + 0.0654i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (-0.608 - 0.793i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + (0.980 + 0.195i)T \)
89 \( 1 + (0.793 + 0.608i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.86297970545424753883132316021, −21.3769362140584633770225015515, −20.263281963899584682837416110683, −19.48927384525803394186702385459, −18.84482130365000127768528825475, −17.74920502518513061563345703288, −17.16214214170946414703855343328, −16.29306495129387469748800858674, −15.893670776248552344124941823, −14.82566804021152795147120759716, −13.84912991503433079321657387763, −12.738518910083196823082479140667, −12.14301238293916088117601474182, −11.52602510640028965571582868709, −10.795960158505687786098838325353, −9.53957241985150993462192149539, −8.98947071543413578449422873327, −7.64774174402338884254967156941, −7.04863241669864654538465950402, −6.05227428943288526201880263890, −5.05928341131405289624233106960, −4.3330751656180911128084789137, −3.498635195260378632563806395926, −1.67121761577061008763992467331, −0.936147970693721789758162447076, 0.35651609236000242007106908838, 1.27499522982823947290368598328, 2.95849515536509368487500211051, 3.798214957028035947712520092810, 4.70668623474086156016433300467, 5.925317223122057947045790730925, 6.376357316961636132198239761991, 7.53560070884817971839597246499, 8.125073553764888086372562164076, 9.5309776187448447148238717012, 10.39727030670938208132396936238, 11.00531344982536533116980139570, 11.96616707320317451326523350776, 12.28204155802120994596862766600, 13.507385346680964308895770417539, 14.60098849952290810439339461846, 15.14791689222240329338055513275, 16.35857044366241102807816777634, 16.50802515840051001596019131643, 17.71915493438061151862336502180, 18.42028277263481700331939695881, 19.00137060182055314806763324239, 19.967765627083013113155951287050, 20.85137658706745137348531289492, 21.86882684742515949225501548288

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