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Properties

Label 1-1311-1311.173-r0-0-0
Degree $1$
Conductor $1311$
Sign $0.000185 - 0.999i$
Analytic cond. $6.08825$
Root an. cond. $6.08825$
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  + (−0.701 + 0.712i)2-s + (−0.0158 − 0.999i)4-s + (−0.296 + 0.954i)5-s + (−0.786 − 0.618i)7-s + (0.723 + 0.690i)8-s + (−0.472 − 0.881i)10-s + (−0.580 + 0.814i)11-s + (0.999 + 0.0317i)13-s + (0.991 − 0.126i)14-s + (−0.999 + 0.0317i)16-s + (−0.873 + 0.486i)17-s + (0.959 + 0.281i)20-s + (−0.173 − 0.984i)22-s + (−0.823 − 0.567i)25-s + (−0.723 + 0.690i)26-s + ⋯
L(s)  = 1  + (−0.701 + 0.712i)2-s + (−0.0158 − 0.999i)4-s + (−0.296 + 0.954i)5-s + (−0.786 − 0.618i)7-s + (0.723 + 0.690i)8-s + (−0.472 − 0.881i)10-s + (−0.580 + 0.814i)11-s + (0.999 + 0.0317i)13-s + (0.991 − 0.126i)14-s + (−0.999 + 0.0317i)16-s + (−0.873 + 0.486i)17-s + (0.959 + 0.281i)20-s + (−0.173 − 0.984i)22-s + (−0.823 − 0.567i)25-s + (−0.723 + 0.690i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000185 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000185 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1311\)    =    \(3 \cdot 19 \cdot 23\)
Sign: $0.000185 - 0.999i$
Analytic conductor: \(6.08825\)
Root analytic conductor: \(6.08825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1311} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1311,\ (0:\ ),\ 0.000185 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05297837997 - 0.05298818653i\)
\(L(\frac12)\) \(\approx\) \(0.05297837997 - 0.05298818653i\)
\(L(1)\) \(\approx\) \(0.4862933017 + 0.2162584883i\)
\(L(1)\) \(\approx\) \(0.4862933017 + 0.2162584883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.701 + 0.712i)T \)
5 \( 1 + (-0.296 + 0.954i)T \)
7 \( 1 + (-0.786 - 0.618i)T \)
11 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (0.999 + 0.0317i)T \)
17 \( 1 + (-0.873 + 0.486i)T \)
29 \( 1 + (0.356 - 0.934i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.841 - 0.540i)T \)
41 \( 1 + (0.678 + 0.734i)T \)
43 \( 1 + (-0.916 + 0.400i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.745 + 0.666i)T \)
59 \( 1 + (-0.204 + 0.978i)T \)
61 \( 1 + (-0.916 - 0.400i)T \)
67 \( 1 + (0.0792 - 0.996i)T \)
71 \( 1 + (-0.266 - 0.963i)T \)
73 \( 1 + (0.630 - 0.776i)T \)
79 \( 1 + (-0.950 + 0.312i)T \)
83 \( 1 + (0.888 - 0.458i)T \)
89 \( 1 + (-0.444 + 0.895i)T \)
97 \( 1 + (0.975 - 0.220i)T \)
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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

−20.894213769449325886115185394870, −20.39988981374743726445865602987, −19.62834495695281410616813917066, −18.835739819915231209647028309682, −18.392977453462571586318694383393, −17.40476891670132427303993734554, −16.57804828185529585492778345725, −15.81605636734133213742715068760, −15.65916641547146244754774401555, −13.85657545732691383275965487487, −13.14118316130108554379719326275, −12.6774173903580102259339324859, −11.74523965729805975537468914882, −11.12623432198673182624858116826, −10.22423073591100226463581181712, −9.21248155292638845195341373339, −8.75493007958332809553598045197, −8.15822243718378033251331856057, −7.052960318235549092652394561101, −6.00976006186746503073290935222, −5.03824614536845969291404643452, −3.91961080796844119365322149625, −3.1795444428250277298076972569, −2.20265807103041963811077598820, −1.03717757143685783765495181291, 0.04284893724189986578613802275, 1.57454939762773486728013956515, 2.689688970792066290849488672395, 3.80286720096596108882810265635, 4.705755367412905544454977888383, 6.057062305238756650608417245455, 6.53517030247912426123882636197, 7.30419199169110371312240254710, 7.974362721132350197431805616387, 8.981573454538160023308150287533, 9.851640415462726518677671303825, 10.644515854244825467045303214409, 10.93235381892874971154168186560, 12.24640790606132166957933988653, 13.39392473921271497242122170500, 13.88775150462638200442772213102, 14.93912868106705157473877355696, 15.55258021018977858881821910284, 16.03892506710313217039310095886, 17.01416430393025711627492174558, 17.867668997587102091444052333579, 18.29556715785139611130896304057, 19.21733895571577273620661681987, 19.73944638135930217802061187433, 20.4809968890547189938044077483

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