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Properties

Label 1-311-311.304-r1-0-0
Degree $1$
Conductor $311$
Sign $0.851 + 0.523i$
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  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (−0.979 + 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (0.954 − 0.299i)17-s + ⋯
L(s)  = 1  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (−0.979 + 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (0.954 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.851 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.107998876 + 1.161637469i\)
\(L(\frac12)\) \(\approx\) \(4.107998876 + 1.161637469i\)
\(L(1)\) \(\approx\) \(2.249043657 + 0.1209511655i\)
\(L(1)\) \(\approx\) \(2.249043657 + 0.1209511655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.918 - 0.394i)T \)
3 \( 1 + (0.918 + 0.394i)T \)
5 \( 1 + (-0.994 - 0.101i)T \)
7 \( 1 + (0.151 + 0.988i)T \)
11 \( 1 + (-0.979 + 0.201i)T \)
13 \( 1 + (0.688 + 0.724i)T \)
17 \( 1 + (0.954 - 0.299i)T \)
19 \( 1 + (0.994 - 0.101i)T \)
23 \( 1 + (-0.347 + 0.937i)T \)
29 \( 1 + (-0.528 + 0.848i)T \)
31 \( 1 + (0.954 + 0.299i)T \)
37 \( 1 + (-0.151 + 0.988i)T \)
41 \( 1 + (0.612 + 0.790i)T \)
43 \( 1 + (-0.151 - 0.988i)T \)
47 \( 1 + (0.528 - 0.848i)T \)
53 \( 1 + (0.151 + 0.988i)T \)
59 \( 1 + (-0.151 - 0.988i)T \)
61 \( 1 + (0.994 - 0.101i)T \)
67 \( 1 + (-0.612 + 0.790i)T \)
71 \( 1 + (0.440 - 0.897i)T \)
73 \( 1 + (-0.758 - 0.651i)T \)
79 \( 1 + (-0.612 - 0.790i)T \)
83 \( 1 + (-0.874 + 0.485i)T \)
89 \( 1 + (0.151 - 0.988i)T \)
97 \( 1 + (0.440 - 0.897i)T \)
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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

−24.61279582249814617703041229609, −24.09562691174323215569576501920, −23.18034024145163503678305233380, −22.71882804633081716927310183737, −20.98808476522811949180385877526, −20.665087499238668007038297538680, −19.80505433250665615935875495260, −18.80279519422420235939203088566, −17.713449143673656833375476657383, −16.29593077099554893207448855723, −15.72003599143966113809087366897, −14.74821803264780598172491369517, −13.97603970954332672330947594198, −13.12698215186813060796654282112, −12.361975084463879033200697608350, −11.18818671527389989952043027115, −10.17412845249798628698142565462, −8.28312788414349295359498508951, −7.84076105526027520359337973824, −7.13593364029925537845648094411, −5.78323334139190302604862523118, −4.33043714963558132359217092579, −3.56420705348617971824577538498, −2.696114221733624493402666822855, −0.88118385641538218658569431614, 1.49788198557222050519121211196, 2.838596029031531403804108852076, 3.51398586070746669723702624811, 4.69624041231699902456108226523, 5.51930012902317221395392225236, 7.17869795304844682872983244298, 8.078261933922759269744099126429, 9.20888973037775272269212109566, 10.27216470580168563798558160332, 11.4538744150055528497578114608, 12.14017620825576670566015008518, 13.22847957667626806218941129247, 14.12178220367459447118543129005, 15.069547665971303436744822179752, 15.751905536847386671860829694656, 16.217935035343960673789082520479, 18.53764669071036153732455374075, 18.87953148265970096320600179123, 19.9921566049864804444191409366, 20.702767189874749651827922359975, 21.399874378060506961796707178291, 22.26256572608714539466427140046, 23.39205773670827330565805103338, 24.02760606368154212278638059946, 25.02724668377474645991954811311

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