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Properties

Label 1-896-896.221-r0-0-0
Degree $1$
Conductor $896$
Sign $0.988 + 0.148i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (−0.321 − 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (0.751 − 0.659i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.258 + 0.965i)17-s + (0.0654 + 0.997i)19-s + (0.608 + 0.793i)23-s + (−0.608 + 0.793i)25-s + (0.831 + 0.555i)27-s + (−0.195 + 0.980i)29-s + (−0.866 + 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.896 − 0.442i)37-s + ⋯
L(s)  = 1  + (−0.321 − 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (0.751 − 0.659i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.258 + 0.965i)17-s + (0.0654 + 0.997i)19-s + (0.608 + 0.793i)23-s + (−0.608 + 0.793i)25-s + (0.831 + 0.555i)27-s + (−0.195 + 0.980i)29-s + (−0.866 + 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.896 − 0.442i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9982584813 + 0.07467720027i\)
\(L(\frac12)\) \(\approx\) \(0.9982584813 + 0.07467720027i\)
\(L(1)\) \(\approx\) \(0.8570935712 - 0.1948196059i\)
\(L(1)\) \(\approx\) \(0.8570935712 - 0.1948196059i\)

Euler product

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

−22.278809465890424829900541823, −21.162176834225967027222058908536, −20.33133541434364858076771447895, −19.79429402705344975791519231268, −18.67133839765102000884614958849, −17.885217748633523910607032227162, −17.21710012594803816369962677623, −16.21314534313136848155729312355, −15.438860265685794465524134195477, −14.958743110223660361856805629702, −14.17783149879235604097945157167, −13.08815992700520703100497670359, −11.92166617293989419970633009397, −11.25418859375061908344823401995, −10.67843015089828777303132934384, −9.70776490518123679754293727204, −9.04464279996163298963510618377, −7.88324809689105264616696366547, −6.88309169336600606176063917028, −6.14883290504927387864870019489, −4.99547538659976213436143317464, −4.18283419519725833553055133925, −3.29773950031324013801454843866, −2.43708896987898241698441175136, −0.52149600379639357672419460428, 1.21720020148213996151227010732, 1.66403890598187647293459306285, 3.34130990080063482738841340708, 4.20259209169668799152043000319, 5.405870064495451214689530132333, 6.13583310851789772194696120122, 7.02873085744653352964523471453, 8.0399957189581021571192725937, 8.663552268364350301380910966598, 9.459974461426536273177958184979, 11.090320478761904886900169724494, 11.359928921248144114964156582443, 12.4959719408956605086322198490, 12.84167855422431775432023123555, 13.88787640071033436914292201992, 14.54384704103241055063103096970, 15.820584112524195075995353203891, 16.690874127175391877974245010916, 16.97081604233036860690182009323, 18.08574923777143782351504949743, 18.89864985386197032466245896399, 19.545039374274479426568922116, 20.14154081144444539217626138156, 21.24599657555790251959529243860, 21.935430788410552872217318993043

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