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Properties

Label 1-896-896.373-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: $\chi_{896} (373, \cdot )$
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

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

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