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Properties

Label 1-896-896.219-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.970 - 0.240i$
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  + (0.442 + 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.659 + 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)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 + (0.321 + 0.946i)37-s + ⋯
L(s)  = 1  + (0.442 + 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.659 + 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)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 + (0.321 + 0.946i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.970 - 0.240i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1815801565 + 1.484889883i\)
\(L(\frac12)\) \(\approx\) \(-0.1815801565 + 1.484889883i\)
\(L(1)\) \(\approx\) \(0.8368162774 + 0.5879749035i\)
\(L(1)\) \(\approx\) \(0.8368162774 + 0.5879749035i\)

Euler product

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

−21.07226558023771587720885097311, −20.48852679241939049293046768461, −19.455564814900958637474305808649, −19.1446048999279219188791492861, −18.42261910717115103129090849570, −17.40341723058120789886488181365, −16.47748426607282287254080237462, −15.82044078878624194407030965164, −14.80965209236385489482155469388, −13.95832861746320526959943609265, −13.39214607576375167662522204244, −12.34604445843119300980014102140, −11.66192202437285975153893962947, −11.16346494207909143033421412905, −9.59384789811174616570099754366, −8.78306607668400960186509648386, −8.16997126975292796270188970864, −7.193876736952505684063357257355, −6.650124756254858975251237898164, −5.38231543285333500828225189442, −4.33117390885718455069458699818, −3.2654210714457951869686673349, −2.48802059201980570679847554052, −1.02160957791816564029845133516, −0.365171110085298547381839382064, 1.351901547944451059820161288901, 2.90210519385013508752666610388, 3.40369701563795133160821324218, 4.4711337988865313055238868843, 5.12705527614914816038360164423, 6.40750910256601454712049900536, 7.64936962915243157114560933046, 8.05541601162359252523384012090, 9.09288035225889534083475536149, 10.19025793438045820668339874600, 10.49246259433177404800216906326, 11.66356397669766434426178185239, 12.36496707149473267458039684700, 13.40433031616962091538124419590, 14.556509266204310888332413046258, 15.04219072240871890076368073269, 15.59202406453830719799496000119, 16.45510798864728829872821476048, 17.33291363467251230723184808941, 18.242786932467819198242922115573, 19.34495035213087862362754838574, 19.83335822647944308004078036596, 20.51182309262479632639076189599, 21.3402491377087125282357217164, 22.24734791806299638931097645915

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