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Properties

Label 1-1127-1127.85-r0-0-0
Degree $1$
Conductor $1127$
Sign $0.375 - 0.926i$
Analytic cond. $5.23376$
Root an. cond. $5.23376$
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  + (0.557 + 0.830i)2-s + (0.794 + 0.607i)3-s + (−0.377 + 0.925i)4-s + (−0.591 − 0.806i)5-s + (−0.0611 + 0.998i)6-s + (−0.979 + 0.202i)8-s + (0.262 + 0.965i)9-s + (0.339 − 0.940i)10-s + (−0.301 − 0.953i)11-s + (−0.862 + 0.505i)12-s + (−0.992 − 0.122i)13-s + (0.0203 − 0.999i)15-s + (−0.714 − 0.699i)16-s + (−0.377 − 0.925i)17-s + (−0.654 + 0.755i)18-s + (−0.959 + 0.281i)19-s + ⋯
L(s)  = 1  + (0.557 + 0.830i)2-s + (0.794 + 0.607i)3-s + (−0.377 + 0.925i)4-s + (−0.591 − 0.806i)5-s + (−0.0611 + 0.998i)6-s + (−0.979 + 0.202i)8-s + (0.262 + 0.965i)9-s + (0.339 − 0.940i)10-s + (−0.301 − 0.953i)11-s + (−0.862 + 0.505i)12-s + (−0.992 − 0.122i)13-s + (0.0203 − 0.999i)15-s + (−0.714 − 0.699i)16-s + (−0.377 − 0.925i)17-s + (−0.654 + 0.755i)18-s + (−0.959 + 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.375 - 0.926i$
Analytic conductor: \(5.23376\)
Root analytic conductor: \(5.23376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1127,\ (0:\ ),\ 0.375 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4526241226 - 0.3051035853i\)
\(L(\frac12)\) \(\approx\) \(0.4526241226 - 0.3051035853i\)
\(L(1)\) \(\approx\) \(0.9875094263 + 0.4598380845i\)
\(L(1)\) \(\approx\) \(0.9875094263 + 0.4598380845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.557 + 0.830i)T \)
3 \( 1 + (0.794 + 0.607i)T \)
5 \( 1 + (-0.591 - 0.806i)T \)
11 \( 1 + (-0.301 - 0.953i)T \)
13 \( 1 + (-0.992 - 0.122i)T \)
17 \( 1 + (-0.377 - 0.925i)T \)
19 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (-0.377 - 0.925i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.182 - 0.983i)T \)
41 \( 1 + (-0.591 - 0.806i)T \)
43 \( 1 + (-0.979 - 0.202i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.101 - 0.994i)T \)
59 \( 1 + (0.339 - 0.940i)T \)
61 \( 1 + (-0.452 + 0.891i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.862 - 0.505i)T \)
73 \( 1 + (0.986 + 0.162i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.262 + 0.965i)T \)
89 \( 1 + (-0.768 + 0.639i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−21.55633331483024080106218186491, −20.50735699633584686294309135605, −19.86050001641812697904660885590, −19.41043660031802459165610930186, −18.63619640300504953009913423930, −18.049709721721421274933970166994, −17.09026900157440110080828270936, −15.45043970438566293279313637337, −14.84916590468115615871691699101, −14.705901395579182348635750112232, −13.48696341836475385408989290882, −12.87105911089168752810111677659, −12.1514084970844935640465759931, −11.42952773470032668609476936364, −10.36286474129283689767933730775, −9.81928522691610698727711235833, −8.7327030227135318125891783459, −7.833133558910483194621021949236, −6.893805163180220114921931180432, −6.28320152554963855542292932881, −4.77795979865708545196103681343, −4.07335273514960712243072570767, −3.094628967311223336451669579578, −2.376882772633660934735017149909, −1.62900404455393091921101713765, 0.15001617567647692044853737754, 2.240690544238124371919807626487, 3.24051176256390046586460685316, 4.02313047869205190575862812329, 4.867242037766508706287770066666, 5.391889080547245435814079379130, 6.72984862423896583676629849248, 7.72544087892575903458985053134, 8.27839392898810339814874885687, 8.95727541432961022413551749740, 9.75727766325221368341240351636, 11.00356999698442373106143928634, 11.956186679827527782408720322627, 12.81758554336387555404619208827, 13.53820829401363997095995330768, 14.23245177265522331384777212166, 15.13349889575245421369365047691, 15.62754609417569748434336947131, 16.47116592821110216934012431391, 16.811903358604799510660992963889, 17.949748117744047156282263252872, 19.13691870956552470670274346386, 19.65905665284632114359801103418, 20.758428097376357762433167750083, 21.11882173148763793134632743590

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