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Properties

Label 1-1152-1152.59-r0-0-0
Degree $1$
Conductor $1152$
Sign $-0.0925 + 0.995i$
Analytic cond. $5.34986$
Root an. cond. $5.34986$
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)5-s + (0.608 + 0.793i)7-s + (−0.997 + 0.0654i)11-s + (−0.659 − 0.751i)13-s + (−0.707 + 0.707i)17-s + (−0.195 − 0.980i)19-s + (0.793 + 0.608i)23-s + (−0.793 + 0.608i)25-s + (0.442 + 0.896i)29-s + (−0.866 + 0.5i)31-s + (0.555 − 0.831i)35-s + (−0.195 + 0.980i)37-s + (0.793 + 0.608i)41-s + (0.0654 + 0.997i)43-s + (−0.258 − 0.965i)47-s + ⋯
L(s)  = 1  + (−0.321 − 0.946i)5-s + (0.608 + 0.793i)7-s + (−0.997 + 0.0654i)11-s + (−0.659 − 0.751i)13-s + (−0.707 + 0.707i)17-s + (−0.195 − 0.980i)19-s + (0.793 + 0.608i)23-s + (−0.793 + 0.608i)25-s + (0.442 + 0.896i)29-s + (−0.866 + 0.5i)31-s + (0.555 − 0.831i)35-s + (−0.195 + 0.980i)37-s + (0.793 + 0.608i)41-s + (0.0654 + 0.997i)43-s + (−0.258 − 0.965i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0925 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0925 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.0925 + 0.995i$
Analytic conductor: \(5.34986\)
Root analytic conductor: \(5.34986\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1152,\ (0:\ ),\ -0.0925 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4906358552 + 0.5383753380i\)
\(L(\frac12)\) \(\approx\) \(0.4906358552 + 0.5383753380i\)
\(L(1)\) \(\approx\) \(0.8413049483 + 0.04003997698i\)
\(L(1)\) \(\approx\) \(0.8413049483 + 0.04003997698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.321 - 0.946i)T \)
7 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (-0.997 + 0.0654i)T \)
13 \( 1 + (-0.659 - 0.751i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (-0.195 - 0.980i)T \)
23 \( 1 + (0.793 + 0.608i)T \)
29 \( 1 + (0.442 + 0.896i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.195 + 0.980i)T \)
41 \( 1 + (0.793 + 0.608i)T \)
43 \( 1 + (0.0654 + 0.997i)T \)
47 \( 1 + (-0.258 - 0.965i)T \)
53 \( 1 + (0.555 + 0.831i)T \)
59 \( 1 + (-0.321 - 0.946i)T \)
61 \( 1 + (-0.896 + 0.442i)T \)
67 \( 1 + (-0.0654 + 0.997i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.946 + 0.321i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−21.04500797298988056648519916135, −20.41321031042178739912620808165, −19.4279433579086811659584581551, −18.76896302489376560564673012290, −18.07604428070011703994176086594, −17.30177778658304223132715492264, −16.43014525580435919583980706054, −15.62640594548288620259296650420, −14.72519034875341548274752507673, −14.18765002069597484588345369808, −13.43772689379790310199809948110, −12.39789769571915910420290397118, −11.456274003541178615535871181533, −10.80875703832573810263153613161, −10.23182257646417828036948949248, −9.19801574193376648147727687905, −8.04331737196935698185258567871, −7.396245658462964039960901089759, −6.8028062459400561386348488869, −5.657071375673728403347100716715, −4.58315764654547841890342764280, −3.90968099905745684305365992808, −2.7083744501028818930186662006, −1.9835601351092769893712384641, −0.29996399679483714275900968034, 1.22735153482293647380960340775, 2.3199926899715329480958670858, 3.23805235638896978632395924728, 4.75147183787985531998920207309, 4.96608479057373216257438655621, 5.907903214939929887176840242933, 7.213667037312905820172783129309, 8.02536768300360815609681561934, 8.694679471283407847775573491227, 9.38095407440547373798755951460, 10.566539267420804656892997586282, 11.25834951118150818386872461666, 12.216121103584794240558953969098, 12.86255131250492346312947143627, 13.41016803376807047480507356998, 14.79688880548222153359354960962, 15.29677600660942258638075538306, 15.92817134187202140307847812268, 16.92534774591439620707799641764, 17.70904232503534317374086427396, 18.22059177364336954012333777917, 19.402625897215374225571311727490, 19.905928692046178587497397341864, 20.73612040841541675076173391682, 21.53168488685763445716134541395

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