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Properties

Label 1-1157-1157.11-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.829 + 0.558i$
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 + (−0.235 + 0.971i)3-s + (−0.235 − 0.971i)4-s + (0.909 + 0.415i)5-s + (0.618 + 0.786i)6-s + (−0.0950 − 0.995i)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 + 12-s + (−0.841 − 0.540i)14-s + (−0.618 + 0.786i)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 + (−0.235 + 0.971i)3-s + (−0.235 − 0.971i)4-s + (0.909 + 0.415i)5-s + (0.618 + 0.786i)6-s + (−0.0950 − 0.995i)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 + 12-s + (−0.841 − 0.540i)14-s + (−0.618 + 0.786i)15-s + (−0.888 + 0.458i)16-s + (0.786 − 0.618i)17-s + (−0.909 + 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.829 + 0.558i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04183427165 + 0.1371759871i\)
\(L(\frac12)\) \(\approx\) \(0.04183427165 + 0.1371759871i\)
\(L(1)\) \(\approx\) \(1.167227493 - 0.2179276749i\)
\(L(1)\) \(\approx\) \(1.167227493 - 0.2179276749i\)

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 \( 1 + (-0.235 + 0.971i)T \)
5 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (-0.0950 - 0.995i)T \)
11 \( 1 + (-0.0950 + 0.995i)T \)
17 \( 1 + (0.786 - 0.618i)T \)
19 \( 1 + (-0.458 + 0.888i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (-0.580 - 0.814i)T \)
31 \( 1 + (0.540 - 0.841i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.690 - 0.723i)T \)
43 \( 1 + (0.580 - 0.814i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.971 + 0.235i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
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 \( 1 + (-0.989 + 0.142i)T \)
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

−21.19334569191925157972175912709, −20.025434614941594028259190573436, −18.90510293678524896160896155102, −18.36390968729955331554701483601, −17.59274948147682149467133415791, −16.83984999702260000012688926714, −16.32951523618905727759609303559, −15.24796425719328123160191045858, −14.333345231534413210687894087649, −13.7905525693950967931825826712, −12.848013713538733928647560735740, −12.58282941254748317039869469934, −11.674652173381780694157912381877, −10.66650216541395765563017665158, −9.21951470094021483237993453965, −8.54326431002440056466730440203, −8.03122861801840206890119311480, −6.561649175336253696455323784821, −6.36674650348054296726734066095, −5.38924239765814054615929695372, −4.95294787062074513859555701828, −3.24429423299818467837310177849, −2.53808034376471383094286633363, −1.418715768214516088875107196262, −0.02202941190411967104770222544, 1.34577202759527737643006680368, 2.36569272439931789834277868153, 3.4259337955324396228778149077, 4.07396931145595227044161573833, 5.01967571433213549443513174334, 5.72224880773801212598406606570, 6.59158501325876182409583178221, 7.70459817140806337519463076903, 9.363691244964603136294906739497, 9.70807605531064550445654781635, 10.38654271100942998491095239619, 10.94726130336922241956072834126, 11.919417468840231656988532898421, 12.76186305616720742875745839989, 13.84459664392556250039127163465, 14.14723365823476895963215722355, 15.05901450523221276658338021405, 15.73895444863075989536687575973, 17.03192184123998203872279018886, 17.33700889098599212550270281574, 18.40195250124569190978002772686, 19.25333037959396354781591451001, 20.28381949881957745174741387562, 20.84012059603064639808982087030, 21.16264661511207372292418373140

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