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Properties

Label 1-1157-1157.332-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.839 - 0.543i$
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  + (0.458 − 0.888i)2-s + (−0.165 − 0.986i)3-s + (−0.580 − 0.814i)4-s + (0.142 + 0.989i)5-s + (−0.952 − 0.304i)6-s + (−0.393 − 0.919i)7-s + (−0.989 + 0.142i)8-s + (−0.945 + 0.327i)9-s + (0.945 + 0.327i)10-s + (0.371 + 0.928i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (0.952 − 0.304i)15-s + (−0.327 + 0.945i)16-s + (0.458 + 0.888i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.165 − 0.986i)3-s + (−0.580 − 0.814i)4-s + (0.142 + 0.989i)5-s + (−0.952 − 0.304i)6-s + (−0.393 − 0.919i)7-s + (−0.989 + 0.142i)8-s + (−0.945 + 0.327i)9-s + (0.945 + 0.327i)10-s + (0.371 + 0.928i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (0.952 − 0.304i)15-s + (−0.327 + 0.945i)16-s + (0.458 + 0.888i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153738090 - 0.3410352966i\)
\(L(\frac12)\) \(\approx\) \(1.153738090 - 0.3410352966i\)
\(L(1)\) \(\approx\) \(0.8909749279 - 0.5404228076i\)
\(L(1)\) \(\approx\) \(0.8909749279 - 0.5404228076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
3 \( 1 + (-0.165 - 0.986i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.393 - 0.919i)T \)
11 \( 1 + (0.371 + 0.928i)T \)
17 \( 1 + (0.458 + 0.888i)T \)
19 \( 1 + (-0.899 - 0.436i)T \)
23 \( 1 + (-0.828 + 0.560i)T \)
29 \( 1 + (0.992 + 0.118i)T \)
31 \( 1 + (0.997 + 0.0713i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.771 + 0.636i)T \)
43 \( 1 + (-0.992 + 0.118i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (-0.986 - 0.165i)T \)
61 \( 1 + (-0.0237 + 0.999i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (0.786 + 0.618i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (-0.371 + 0.928i)T \)
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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

−21.386923581322619272512026292599, −21.03866064459404061381351503510, −20.00250290204802943921200276394, −18.96308856316344608787614750000, −18.01195276487966983976195057552, −17.04719364132220910196771015521, −16.56365384906075340136028584218, −15.880702773940079254873057375517, −15.47663934061835646250817075275, −14.30953319254598261980584233289, −13.89078030220558041895133413685, −12.65418989722559792360182048223, −12.148970933934699157939801884236, −11.3216932783280076882678972062, −9.906580596320522031157531550893, −9.30210193999698882498809215464, −8.52468919947063498092759662925, −8.07002297080581660125488787035, −6.27791708155192089926022016907, −6.0240916914877748880921023405, −5.06198432294420257628985898700, −4.42582601198547874921755659369, −3.48433018397280852824031290833, −2.53386355833204563111533527479, −0.48224198689785296027908926786, 1.09274655155983474744854412812, 2.00861145530179158550255152864, 2.848942241296128249800725371394, 3.793265149009314207436759921757, 4.68969887097024594156714774141, 6.127169419177620912842322540471, 6.45302992162682667281161259413, 7.40458320639101747830288915077, 8.356689768615868055220550910538, 9.72902564897469149781622048486, 10.266739102886646248683795438779, 11.05286821097417718833314583634, 11.83798628650890396738040546227, 12.62273516373559392225103881368, 13.31067479665986378010683538287, 14.03509136976575145512528675866, 14.61426224142797095255973534382, 15.46474173919046788450656058807, 16.998197654556605381265331498307, 17.56433487195968156573828697344, 18.25657195553968557334945742535, 19.09932259829934652377903655712, 19.70990001666017339101262975550, 20.08774963441885371447086756061, 21.3607260781618368286393468111

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