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Properties

Label 1-1157-1157.1003-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.166 - 0.986i$
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  + (0.998 − 0.0475i)2-s + (0.771 − 0.636i)3-s + (0.995 − 0.0950i)4-s + (−0.142 − 0.989i)5-s + (0.739 − 0.672i)6-s + (0.992 + 0.118i)7-s + (0.989 − 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (−0.618 + 0.786i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (−0.739 − 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.998 − 0.0475i)2-s + (0.771 − 0.636i)3-s + (0.995 − 0.0950i)4-s + (−0.142 − 0.989i)5-s + (0.739 − 0.672i)6-s + (0.992 + 0.118i)7-s + (0.989 − 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (−0.618 + 0.786i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (−0.739 − 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.166 - 0.986i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.166 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.187502474 - 2.694971060i\)
\(L(\frac12)\) \(\approx\) \(3.187502474 - 2.694971060i\)
\(L(1)\) \(\approx\) \(2.365807573 - 1.059926578i\)
\(L(1)\) \(\approx\) \(2.365807573 - 1.059926578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.998 - 0.0475i)T \)
3 \( 1 + (0.771 - 0.636i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (0.992 + 0.118i)T \)
11 \( 1 + (-0.618 + 0.786i)T \)
17 \( 1 + (-0.998 - 0.0475i)T \)
19 \( 1 + (0.828 - 0.560i)T \)
23 \( 1 + (-0.899 - 0.436i)T \)
29 \( 1 + (0.393 + 0.919i)T \)
31 \( 1 + (0.997 + 0.0713i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.165 - 0.986i)T \)
43 \( 1 + (-0.393 + 0.919i)T \)
47 \( 1 + (-0.415 + 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.636 - 0.771i)T \)
61 \( 1 + (0.853 + 0.520i)T \)
67 \( 1 + (-0.0950 + 0.995i)T \)
71 \( 1 + (0.928 - 0.371i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (0.618 + 0.786i)T \)
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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

−21.52723723852186481588565540336, −20.84682629029189376187751457391, −20.157489687143358033066146350166, −19.36633800600087406918555259708, −18.52971069509953501355040001029, −17.573342028192631835466721944405, −16.457695951568783633030834354895, −15.5674554212940638615304079849, −15.27275331246970580411969035161, −14.33112311602095178829707179184, −13.81509868152090245373895214537, −13.34862232807221073814928322190, −11.77405135212336576598899140122, −11.34705917250915719792420933566, −10.4795616349240006020422454792, −9.90210878570074362660945648578, −8.23172875178110413810861672639, −8.0011093233511020476724970232, −6.95145373998572353375076603394, −5.921754816165148868384687315011, −5.00201549750967899631350750486, −4.16886274920153632481928073152, −3.37223237719324000457393358318, −2.59946649975133055660275199483, −1.77860721321209279799373512257, 1.14172740234286196243812391005, 2.00507788330288536239008160443, 2.717735426145848831356877645166, 4.04252633996518512575895155007, 4.69300627167751908126635971042, 5.43739034746636370694175664750, 6.62385606365950161801272214920, 7.50494457570335773094057608875, 8.12265965105809801406627825098, 8.959452273403170489723569827580, 10.03414799291973855258528378392, 11.21318396353332284687644288023, 11.99035431601233541208305302252, 12.627132737364234084111854913668, 13.25610382193458213947110657000, 14.0579725786311850032507120609, 14.64732181473294215410090562145, 15.70484449425924841081407973030, 15.93400063881718107543237471060, 17.52111205017963935070262382678, 17.816379348707514173959357709780, 19.04811293220779872063979675518, 20.10774287537061410919369737602, 20.2322411321952008253920335060, 20.978248436952195821982109646632

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