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Properties

Label 1-1127-1127.463-r0-0-0
Degree $1$
Conductor $1127$
Sign $0.282 - 0.959i$
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  + (−0.377 + 0.925i)2-s + (0.262 + 0.965i)3-s + (−0.714 − 0.699i)4-s + (−0.301 + 0.953i)5-s + (−0.992 − 0.122i)6-s + (0.917 − 0.396i)8-s + (−0.862 + 0.505i)9-s + (−0.768 − 0.639i)10-s + (−0.818 + 0.574i)11-s + (0.488 − 0.872i)12-s + (0.970 + 0.242i)13-s + (−0.999 − 0.0407i)15-s + (0.0203 + 0.999i)16-s + (−0.714 + 0.699i)17-s + (−0.142 − 0.989i)18-s + (0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (−0.377 + 0.925i)2-s + (0.262 + 0.965i)3-s + (−0.714 − 0.699i)4-s + (−0.301 + 0.953i)5-s + (−0.992 − 0.122i)6-s + (0.917 − 0.396i)8-s + (−0.862 + 0.505i)9-s + (−0.768 − 0.639i)10-s + (−0.818 + 0.574i)11-s + (0.488 − 0.872i)12-s + (0.970 + 0.242i)13-s + (−0.999 − 0.0407i)15-s + (0.0203 + 0.999i)16-s + (−0.714 + 0.699i)17-s + (−0.142 − 0.989i)18-s + (0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.282 - 0.959i$
Analytic conductor: \(5.23376\)
Root analytic conductor: \(5.23376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1127,\ (0:\ ),\ 0.282 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3885651168 + 0.2905400647i\)
\(L(\frac12)\) \(\approx\) \(-0.3885651168 + 0.2905400647i\)
\(L(1)\) \(\approx\) \(0.3490523563 + 0.5942023965i\)
\(L(1)\) \(\approx\) \(0.3490523563 + 0.5942023965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.377 + 0.925i)T \)
3 \( 1 + (0.262 + 0.965i)T \)
5 \( 1 + (-0.301 + 0.953i)T \)
11 \( 1 + (-0.818 + 0.574i)T \)
13 \( 1 + (0.970 + 0.242i)T \)
17 \( 1 + (-0.714 + 0.699i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.714 + 0.699i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.933 - 0.359i)T \)
41 \( 1 + (-0.301 + 0.953i)T \)
43 \( 1 + (0.917 + 0.396i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.979 - 0.202i)T \)
59 \( 1 + (-0.768 - 0.639i)T \)
61 \( 1 + (-0.591 - 0.806i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.488 + 0.872i)T \)
73 \( 1 + (0.947 + 0.320i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (-0.862 + 0.505i)T \)
89 \( 1 + (0.182 - 0.983i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−20.712464453765868804843024678848, −20.0510944843599713068319267995, −19.107702668765790333019126322516, −18.5862965472329246016431874464, −17.91455876423473109097537589019, −17.09225740789861992941421438299, −16.24329651283499414708662630359, −15.42942745134409036222966445597, −13.79694167486754973916200647526, −13.60384783174529129639067904217, −12.796991060294884444287038217226, −12.09054615486442600553340423956, −11.375589544379740858102355960952, −10.61338279757106217394644166465, −9.22494051018331537302267269347, −8.89456237953193167987742332170, −7.85884791989122025161016223391, −7.560078122864468973401075968414, −5.97839947809606563417844976305, −5.168268746452336710151188800108, −3.94067908418559926318332983726, −3.12893820326869373385876262324, −2.09405164692729080518232553422, −1.16922675771244702304806897109, −0.23611456037401274104812417719, 1.8424937268946126685251202242, 3.147968834112594652443531570267, 3.976037576315628380587511394919, 4.909943142176959567463354281869, 5.756064650057587907942554394390, 6.72708671733063415486232981673, 7.52931394367981041708755026374, 8.37512750491159094438857996269, 9.15407982173496271392181584592, 9.98021618716280514124152770390, 10.81580905712541301430656137852, 11.15561415057075564961115921712, 12.78791628493509883268459184078, 13.80241249189844571894312049941, 14.35310877669939302722892583670, 15.29334802381191125436889032467, 15.602624805799831852963186956887, 16.25637606447792802032877205880, 17.297000976629645546445404322927, 18.08043757812945927332803919001, 18.62852367330984115551308057447, 19.66612834695515839331164529665, 20.23231759812863859061424379249, 21.364060796210132855545021321010, 22.09947367198221554074651034991

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