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Properties

Label 1-1157-1157.566-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.596 - 0.802i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.971 + 0.235i)2-s + (−0.888 − 0.458i)3-s + (0.888 − 0.458i)4-s + (0.755 − 0.654i)5-s + (0.971 + 0.235i)6-s + (−0.189 − 0.981i)7-s + (−0.755 + 0.654i)8-s + (0.580 + 0.814i)9-s + (−0.580 + 0.814i)10-s + (0.189 − 0.981i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.971 + 0.235i)15-s + (0.580 − 0.814i)16-s + (−0.235 + 0.971i)17-s + (−0.755 − 0.654i)18-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s + (−0.888 − 0.458i)3-s + (0.888 − 0.458i)4-s + (0.755 − 0.654i)5-s + (0.971 + 0.235i)6-s + (−0.189 − 0.981i)7-s + (−0.755 + 0.654i)8-s + (0.580 + 0.814i)9-s + (−0.580 + 0.814i)10-s + (0.189 − 0.981i)11-s − 12-s + (0.415 + 0.909i)14-s + (−0.971 + 0.235i)15-s + (0.580 − 0.814i)16-s + (−0.235 + 0.971i)17-s + (−0.755 − 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9293637127 - 0.4674033631i\)
\(L(\frac12)\) \(\approx\) \(0.9293637127 - 0.4674033631i\)
\(L(1)\) \(\approx\) \(0.6083499361 - 0.1675392802i\)
\(L(1)\) \(\approx\) \(0.6083499361 - 0.1675392802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.971 + 0.235i)T \)
3 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (-0.189 - 0.981i)T \)
11 \( 1 + (0.189 - 0.981i)T \)
17 \( 1 + (-0.235 + 0.971i)T \)
19 \( 1 + (-0.814 + 0.580i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (-0.327 + 0.945i)T \)
31 \( 1 + (-0.909 + 0.415i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.998 + 0.0475i)T \)
43 \( 1 + (0.327 + 0.945i)T \)
47 \( 1 + (0.540 - 0.841i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (0.458 + 0.888i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (0.945 - 0.327i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (-0.281 - 0.959i)T \)
97 \( 1 + (0.189 + 0.981i)T \)
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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

−21.05297310818561716808596846625, −20.70797061920726750442561216424, −19.39965133380680578968679891351, −18.66066037650615116517689487857, −18.06602036744831802582008630445, −17.39674483885526254931612707108, −16.93477021327602252599340542080, −15.7180287068631561794239508903, −15.37296883135955546326478200541, −14.51682761697063777161358911769, −12.98885755353790394945560350023, −12.41192214631813643987429782745, −11.41302442421430567958414408600, −10.97227194878588699211021324174, −10.00974921145365609039749542729, −9.39480721501788601504023032939, −8.915292467467647579475163846191, −7.307843489538891021410528554423, −6.80095034089146284838323483265, −5.93618026777418500818382752467, −5.140021603004477519174187508743, −3.83678868621195109455503537719, −2.57641841018501453001433896178, −2.00939314795011508676483287247, −0.58268460244372295648058207523, 0.60410632188806741018046713222, 1.24516977860443469175583894087, 2.107915482014514427690905836261, 3.6711919237416900216527716461, 4.99241533712948013111824590852, 5.83082056269535126141181424743, 6.483625233374589915942134242187, 7.2012364126179947916485671226, 8.26662524069928807699326726344, 8.917937593649906576498857089231, 10.01072871203191568575597290601, 10.70683543188581689156505393469, 11.12726350272607701348701106345, 12.39662011692151454414948408072, 12.991730513144550632425592165944, 13.88137597499096318788047056391, 14.81901141959773329707029077740, 16.13834144478688687605574002809, 16.651054683862484745398432218381, 17.03104400017985507982211138225, 17.712550230406208883808478506894, 18.51978200466955548395160865705, 19.35763547675022039249688490261, 19.89860833444156650714174888517, 21.04637918934778096985938879211

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