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Properties

Label 1-896-896.275-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.597 - 0.801i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.597 - 0.801i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.597 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5628754303 - 1.121135751i\)
\(L(\frac12)\) \(\approx\) \(0.5628754303 - 1.121135751i\)
\(L(1)\) \(\approx\) \(1.172556288 - 0.1105551772i\)
\(L(1)\) \(\approx\) \(1.172556288 - 0.1105551772i\)

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

−21.99075004413941570655884368671, −20.94503334141640818204142769280, −20.4014010423766350997147417686, −19.78203178819395476087672691899, −19.00040870150364726647580845437, −18.39146469175636990816310313200, −17.00103700897352633896074287231, −16.46137255283509497187190656265, −15.52547782161357035898751895675, −14.86074415071255544569055367872, −14.08972518298723869475365888153, −13.225679121132002530133008363235, −12.365816388143115938637590097268, −11.68562696931768709062722533306, −10.59222492244841226860853857397, −9.44651281911553578853962935739, −8.888602921667803020382092660393, −8.22489002616482472204224308459, −7.20116507849666289018849423837, −6.50399600703344696652183641351, −4.89982864046038577038629710257, −4.142915375316677993559794902121, −3.57343094664683575430108807577, −2.137001336344580236152034957177, −1.36525749810326658465355776010, 0.22206212792226441139744137888, 1.62695202797240542215241867852, 2.760900383630968476048010919123, 3.58255329264565009288466652192, 4.17935666554261365918108143598, 5.59151264648481329262461827772, 6.898041289798768209651581758670, 7.279212301689155598900962438526, 8.4546897686870831897605773962, 8.89179843594037965278589850265, 10.0132726108208469494228426000, 10.96252783851811059543325622044, 11.61463832191747076492511354898, 12.82653750738650312233823610587, 13.439432347772255870371268539883, 14.38247196650002137340543575828, 15.180381594759833225291267443548, 15.47969857610018482654826124448, 16.58296214801436715429836166091, 17.742869024857428513263692878055, 18.40634720865722181802484443352, 19.41083462974338165595780550337, 19.74169646470755365108114284737, 20.404699691313643089280909273283, 21.621772547531788363085831755906

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