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Properties

Label 1-896-896.187-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.453 - 0.891i$
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  + (0.896 − 0.442i)3-s + (−0.321 − 0.946i)5-s + (0.608 − 0.793i)9-s + (0.997 − 0.0654i)11-s + (−0.980 + 0.195i)13-s + (−0.707 − 0.707i)15-s + (−0.258 − 0.965i)17-s + (0.751 − 0.659i)19-s + (−0.793 − 0.608i)23-s + (−0.793 + 0.608i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.946 − 0.321i)37-s + ⋯
L(s)  = 1  + (0.896 − 0.442i)3-s + (−0.321 − 0.946i)5-s + (0.608 − 0.793i)9-s + (0.997 − 0.0654i)11-s + (−0.980 + 0.195i)13-s + (−0.707 − 0.707i)15-s + (−0.258 − 0.965i)17-s + (0.751 − 0.659i)19-s + (−0.793 − 0.608i)23-s + (−0.793 + 0.608i)25-s + (0.195 − 0.980i)27-s + (−0.555 + 0.831i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.946 − 0.321i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9129736733 - 1.488372595i\)
\(L(\frac12)\) \(\approx\) \(0.9129736733 - 1.488372595i\)
\(L(1)\) \(\approx\) \(1.176187433 - 0.6001909606i\)
\(L(1)\) \(\approx\) \(1.176187433 - 0.6001909606i\)

Euler product

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

−21.91656643303109292549144113633, −21.78258415050794467913698605705, −20.4001085536796208518773245033, −19.82538973612966088529081999713, −19.22653307686655132528355079494, −18.457121784875561520941549271178, −17.44093328456202275519629751039, −16.5766806047007425137435357995, −15.55445874993256077993080075480, −14.936579315058787597645062493766, −14.35406657670941333869555258071, −13.68557783912899730428406394409, −12.50180417875649671020008014650, −11.64989156785912219544126611594, −10.68614600817130885387362027467, −9.88422680884635119630298972816, −9.2798801519855605379233119705, −8.06904714226111297460593300086, −7.52455529285824206962708890763, −6.57668465783337490803873751499, −5.464037124084957714028005729580, −4.0769173446404392430064492138, −3.68076853695404825097664274292, −2.589878909594476511152794189780, −1.69177693738792139752344143051, 0.67946392757859428131702507091, 1.785916713021621649328764833363, 2.79410541902984383895881094734, 3.94357429082151604838587189775, 4.64560823034904851756116760916, 5.81322873558238072995217420186, 7.154513480075828724598478033250, 7.49129668727141712936146355205, 8.77530609696549698935734181977, 9.1341922619305730332148408910, 9.923872448931272018926292532671, 11.48276386713637269233429822188, 12.079061983144089845239286034033, 12.83483799658988673512916067177, 13.69813411878480709092006253955, 14.4023236223653822862096890038, 15.19515040413258538714213795578, 16.16024973917070653439052592132, 16.83258134244337747846415469583, 17.82808476688198977342322801453, 18.630589217083300342526157912638, 19.58408847404868994396401885484, 20.15029110337090278499183996577, 20.46991438711940958923133462866, 21.726200516843793375138268010889

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