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Properties

Label 1-1152-1152.5-r1-0-0
Degree $1$
Conductor $1152$
Sign $0.995 + 0.0925i$
Analytic cond. $123.799$
Root an. cond. $123.799$
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

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

Invariants

Degree: \(1\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.995 + 0.0925i$
Analytic conductor: \(123.799\)
Root analytic conductor: \(123.799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1152,\ (1:\ ),\ 0.995 + 0.0925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.493055455 + 0.06926814377i\)
\(L(\frac12)\) \(\approx\) \(1.493055455 + 0.06926814377i\)
\(L(1)\) \(\approx\) \(0.9456377813 + 0.07679883238i\)
\(L(1)\) \(\approx\) \(0.9456377813 + 0.07679883238i\)

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

−20.96970618083932306813817028681, −20.43360332157571972399516562760, −19.66582869755793530730701181201, −18.5871406187651675860658085701, −18.13289530512571671130103145998, −17.25184135641774577798495359679, −16.17612958922287224414415520243, −15.8651212760755734045953194330, −15.10542921592960705961866594944, −13.74701879328212019815015097726, −13.30137474328167040237209688639, −12.45520431725044625215541231085, −11.918087235443728466556878191074, −10.66594853747016785307228009950, −9.96492947399022628397733227162, −9.04372371126778697072419200213, −8.437877769567869838641080087149, −7.60340613742279834864716658003, −6.291723884147821528847373356860, −5.610486536333858577448080096031, −4.97933490321656520034666763070, −3.760659207075652925921034288411, −2.756442934448900861260798244458, −1.81086699858718555429098794462, −0.56984689990087619434861266833, 0.499928234384381625047680073, 2.029678047685953059811644735435, 2.76043311997153551341348825954, 3.85152305441521588998985167527, 4.570174804450955801756235389338, 6.089712425966474548886094255026, 6.417801153861339898604041022308, 7.38807682012292795198223291497, 8.18306217068952230840549432821, 9.39188351819101395463056731202, 10.05218905160510458068863788262, 10.92733099860704887346780827215, 11.29029889624251635796333491291, 12.76700984066080082656603264326, 13.33642366565782197270067984120, 13.98798556096702752070147901117, 14.872133932496965098642971960503, 15.75983600445256297967814692166, 16.32317546724888693970344732777, 17.51399675956613515340726470978, 17.86466034428397524826892891031, 18.99549153288121331986857196945, 19.3146794648583640218527717223, 20.41051518075460096515272552186, 21.125356244685935555967113062579

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