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Properties

Label 1-1157-1157.1073-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.988 + 0.150i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.618 − 0.786i)2-s + (−0.971 + 0.235i)3-s + (−0.235 + 0.971i)4-s + (0.909 − 0.415i)5-s + (0.786 + 0.618i)6-s + (−0.995 − 0.0950i)7-s + (0.909 − 0.415i)8-s + (0.888 − 0.458i)9-s + (−0.888 − 0.458i)10-s + (0.0950 + 0.995i)11-s − i·12-s + (0.540 + 0.841i)14-s + (−0.786 + 0.618i)15-s + (−0.888 − 0.458i)16-s + (−0.786 − 0.618i)17-s + (−0.909 − 0.415i)18-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.971 + 0.235i)3-s + (−0.235 + 0.971i)4-s + (0.909 − 0.415i)5-s + (0.786 + 0.618i)6-s + (−0.995 − 0.0950i)7-s + (0.909 − 0.415i)8-s + (0.888 − 0.458i)9-s + (−0.888 − 0.458i)10-s + (0.0950 + 0.995i)11-s − i·12-s + (0.540 + 0.841i)14-s + (−0.786 + 0.618i)15-s + (−0.888 − 0.458i)16-s + (−0.786 − 0.618i)17-s + (−0.909 − 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5552762898 + 0.04198887363i\)
\(L(\frac12)\) \(\approx\) \(0.5552762898 + 0.04198887363i\)
\(L(1)\) \(\approx\) \(0.5092313458 - 0.1450538712i\)
\(L(1)\) \(\approx\) \(0.5092313458 - 0.1450538712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.618 - 0.786i)T \)
3 \( 1 + (-0.971 + 0.235i)T \)
5 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (-0.995 - 0.0950i)T \)
11 \( 1 + (0.0950 + 0.995i)T \)
17 \( 1 + (-0.786 - 0.618i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (-0.998 - 0.0475i)T \)
29 \( 1 + (-0.814 - 0.580i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.723 - 0.690i)T \)
43 \( 1 + (0.814 - 0.580i)T \)
47 \( 1 + (-0.281 - 0.959i)T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (0.189 + 0.981i)T \)
67 \( 1 + (-0.971 + 0.235i)T \)
71 \( 1 + (0.814 - 0.580i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.0950 - 0.995i)T \)
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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

−21.49308833644534852914948434197, −19.94043944439415478655953650157, −19.15518654215827888097197934882, −18.63793835949206244274442154972, −17.78254084515946376872552584081, −17.29550042352352952476944022469, −16.47470127418978574013169747286, −15.977558863862208504496428186940, −15.05849264723875109604577033012, −14.03712271597864166375762508557, −13.28726629376346719744123738440, −12.68295141724963509869406809162, −11.23613750970262423931238832904, −10.71882916322314594951876018912, −9.947485198534627058882567690021, −9.209194227807376701330282796832, −8.27423676213816187588668673568, −7.087948814930257167398858228444, −6.27461508371324805930138140131, −6.16025411429134687022277983062, −5.20824718027456634299389257368, −4.02404022062658224615400436086, −2.50336950743793867034353682958, −1.45431148759889016658365262470, −0.28684694512078882021533591584, 0.543932592536826305541102625909, 1.75273072033013290750153624134, 2.52142634240899820505230039273, 3.99159837625505060125972382559, 4.5380942979564574665611109233, 5.75819179609619813739669977840, 6.56552529974925591509551499212, 7.359816051022173148795941179685, 8.72310376428938042594579253492, 9.51247930192885296008285277565, 10.08354696608489734386006266667, 10.53024779122811797791589247936, 11.80832901271630850930177688064, 12.24203785583681631263268283623, 13.15321048303561973743479408309, 13.54843302966800033401636048235, 15.13999242077267445949799690190, 16.024317251477315119151313759924, 16.84820837912730545377457841439, 17.18224444374880960719139341028, 18.05131763891672335366277189791, 18.583099932628672161434702091212, 19.619151822284856477205464039064, 20.46908974122040735191345302925, 20.9725947873092037022119720496

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