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Properties

Label 1-1157-1157.736-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.994 - 0.100i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.540 + 0.841i)2-s + (−0.936 − 0.349i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (−0.212 − 0.977i)6-s + (0.599 − 0.800i)7-s + (−0.989 + 0.142i)8-s + (0.755 + 0.654i)9-s + (−0.755 + 0.654i)10-s + (−0.989 − 0.142i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (0.212 − 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (−0.936 − 0.349i)3-s + (−0.415 + 0.909i)4-s + (0.142 + 0.989i)5-s + (−0.212 − 0.977i)6-s + (0.599 − 0.800i)7-s + (−0.989 + 0.142i)8-s + (0.755 + 0.654i)9-s + (−0.755 + 0.654i)10-s + (−0.989 − 0.142i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (0.212 − 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (736, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.994 - 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053026548 - 0.05329892421i\)
\(L(\frac12)\) \(\approx\) \(1.053026548 - 0.05329892421i\)
\(L(1)\) \(\approx\) \(0.8997256195 + 0.3098415341i\)
\(L(1)\) \(\approx\) \(0.8997256195 + 0.3098415341i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (-0.936 - 0.349i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.599 - 0.800i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.0713 - 0.997i)T \)
23 \( 1 + (0.0713 + 0.997i)T \)
29 \( 1 + (0.599 - 0.800i)T \)
31 \( 1 + (-0.997 - 0.0713i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.936 - 0.349i)T \)
43 \( 1 + (-0.599 - 0.800i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (-0.349 - 0.936i)T \)
61 \( 1 + (-0.877 + 0.479i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (0.212 + 0.977i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−21.3758646159174807039686417182, −20.772152195030471847130973287828, −20.09360367672102400624448812739, −18.8389182319184837178226196373, −18.29990871007629599103128021070, −17.59358213801507023608304205625, −16.5775886783522881127193235451, −15.90932877964790816196608697182, −14.992730352307376020680268622922, −14.339672498218323719866093997630, −12.94793995829191318301527401184, −12.60535002219302240516128538441, −12.031958621151488802923542104483, −11.097525340125402522306231505684, −10.40983876575080109662423628682, −9.65378374143710423295362133123, −8.74015342818380256322436644241, −7.86593104793759638560326905680, −6.13045675879561898188736067319, −5.71748943782969305004370569514, −4.83497916888804350540109367786, −4.42873276606810758843948742121, −3.15190850819260268492597217587, −1.89159992620395265111041629676, −1.12451624284665544716636415848, 0.448979037973785525092834330606, 2.15893448902128307676356528515, 3.23562468960369478343155560969, 4.338449239699139931003170284837, 5.19526165233077881051756460266, 5.835635024550580311114716690458, 6.83233869128615778177795508088, 7.47351552571307498045573364152, 7.836833119917018843471279809941, 9.38367233662541675331196639404, 10.430706609474299589227418886311, 11.15408739784857991590860433903, 11.774332681061394952572272544008, 12.88579484595592108912982847576, 13.64379328662426797183130035492, 14.06225028808360488328968738506, 15.20117454508226883368406198430, 15.77471454094759580906362208552, 16.68188961631710197645351098828, 17.40191859861496023613261061927, 18.058774676426898305377953247237, 18.460078404512325974374413169541, 19.6015411175988196241197230215, 20.91754280195255615214478064235, 21.50062827694830294028328504428

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