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Properties

Label 1-311-311.262-r1-0-0
Degree $1$
Conductor $311$
Sign $0.0565 + 0.998i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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  + (0.688 − 0.724i)2-s + (0.688 + 0.724i)3-s + (−0.0506 − 0.998i)4-s + (0.979 + 0.201i)5-s + 6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.0506 + 0.998i)9-s + (0.820 − 0.571i)10-s + (−0.918 + 0.394i)11-s + (0.688 − 0.724i)12-s + (−0.0506 + 0.998i)13-s + (−0.440 + 0.897i)14-s + (0.528 + 0.848i)15-s + (−0.994 + 0.101i)16-s + (−0.820 + 0.571i)17-s + ⋯
L(s)  = 1  + (0.688 − 0.724i)2-s + (0.688 + 0.724i)3-s + (−0.0506 − 0.998i)4-s + (0.979 + 0.201i)5-s + 6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.0506 + 0.998i)9-s + (0.820 − 0.571i)10-s + (−0.918 + 0.394i)11-s + (0.688 − 0.724i)12-s + (−0.0506 + 0.998i)13-s + (−0.440 + 0.897i)14-s + (0.528 + 0.848i)15-s + (−0.994 + 0.101i)16-s + (−0.820 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0565 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0565 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.0565 + 0.998i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.0565 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.742046027 + 1.646093137i\)
\(L(\frac12)\) \(\approx\) \(1.742046027 + 1.646093137i\)
\(L(1)\) \(\approx\) \(1.618269421 + 0.1342370709i\)
\(L(1)\) \(\approx\) \(1.618269421 + 0.1342370709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.688 - 0.724i)T \)
3 \( 1 + (0.688 + 0.724i)T \)
5 \( 1 + (0.979 + 0.201i)T \)
7 \( 1 + (-0.954 + 0.299i)T \)
11 \( 1 + (-0.918 + 0.394i)T \)
13 \( 1 + (-0.0506 + 0.998i)T \)
17 \( 1 + (-0.820 + 0.571i)T \)
19 \( 1 + (-0.979 + 0.201i)T \)
23 \( 1 + (0.758 + 0.651i)T \)
29 \( 1 + (0.440 + 0.897i)T \)
31 \( 1 + (-0.820 - 0.571i)T \)
37 \( 1 + (0.954 + 0.299i)T \)
41 \( 1 + (0.250 - 0.968i)T \)
43 \( 1 + (0.954 - 0.299i)T \)
47 \( 1 + (-0.440 - 0.897i)T \)
53 \( 1 + (-0.954 + 0.299i)T \)
59 \( 1 + (0.954 - 0.299i)T \)
61 \( 1 + (-0.979 + 0.201i)T \)
67 \( 1 + (-0.250 - 0.968i)T \)
71 \( 1 + (0.612 + 0.790i)T \)
73 \( 1 + (0.151 + 0.988i)T \)
79 \( 1 + (-0.250 + 0.968i)T \)
83 \( 1 + (0.528 - 0.848i)T \)
89 \( 1 + (-0.954 - 0.299i)T \)
97 \( 1 + (0.612 + 0.790i)T \)
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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

−25.0065777590879546854855079667, −24.02384768029997456525858394251, −23.157081867415976125193182975785, −22.307320773987680308402264032458, −21.196212184256308521265320554683, −20.5374003592825538586282819254, −19.47699880383206155646873197887, −18.22361307298582897255762241134, −17.608696780846954659295963062279, −16.52568233638524356351969505019, −15.569761431783170687414038749742, −14.59933302736584569236632740885, −13.608411348566470317852668629519, −12.96015177499090470442834462943, −12.69627254378958700875274567069, −10.869349825421813257728536120711, −9.515464747843062295575264021768, −8.61483785769817087095180149403, −7.60923497851743271763428751368, −6.54105695050760688167022215084, −5.92389539578263749707110965303, −4.59942379861015539919293699007, −3.03562809027860185334098058077, −2.493704908930935231058952097087, −0.44843736718068453478041273626, 1.96243565810854238920017463820, 2.58981742717746583055796997469, 3.73014549080963474174333725600, 4.82334949175037968257293536861, 5.855007488480452695771818227875, 6.921994975670439118935218384333, 8.86779888742186395016469732364, 9.483607389169719379053469567157, 10.34836985893749565727045098954, 11.05541431273809981275024392017, 12.743880301710076496302070487960, 13.196961095466930293760737073281, 14.16162917308624259811986430153, 15.02704431242369889531943452723, 15.80366866575045991867156431656, 16.962221683820162329120184185204, 18.40911015372489161919837836478, 19.15830397654806744552756245977, 20.0191104622566623204496787005, 21.01543269885080404082707719866, 21.6008995430054450164756538094, 22.16099523940121868639980445709, 23.20476939990393420543692475772, 24.274687866973200706103686710359, 25.55968112700289672600981722261

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