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Properties

Label 1-311-311.68-r1-0-0
Degree $1$
Conductor $311$
Sign $0.971 + 0.236i$
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  + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (−0.820 − 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (0.612 + 0.790i)17-s + ⋯
L(s)  = 1  + (0.347 + 0.937i)2-s + (0.347 − 0.937i)3-s + (−0.758 + 0.651i)4-s + (−0.954 + 0.299i)5-s + 6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.758 − 0.651i)9-s + (−0.612 − 0.790i)10-s + (−0.820 − 0.571i)11-s + (0.347 + 0.937i)12-s + (−0.758 − 0.651i)13-s + (−0.994 − 0.101i)14-s + (−0.0506 + 0.998i)15-s + (0.151 − 0.988i)16-s + (0.612 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.971 + 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328822652 + 0.1595775922i\)
\(L(\frac12)\) \(\approx\) \(1.328822652 + 0.1595775922i\)
\(L(1)\) \(\approx\) \(0.9304564387 + 0.2501864737i\)
\(L(1)\) \(\approx\) \(0.9304564387 + 0.2501864737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.347 + 0.937i)T \)
3 \( 1 + (0.347 - 0.937i)T \)
5 \( 1 + (-0.954 + 0.299i)T \)
7 \( 1 + (-0.440 + 0.897i)T \)
11 \( 1 + (-0.820 - 0.571i)T \)
13 \( 1 + (-0.758 - 0.651i)T \)
17 \( 1 + (0.612 + 0.790i)T \)
19 \( 1 + (0.954 + 0.299i)T \)
23 \( 1 + (0.874 + 0.485i)T \)
29 \( 1 + (0.994 - 0.101i)T \)
31 \( 1 + (0.612 - 0.790i)T \)
37 \( 1 + (0.440 + 0.897i)T \)
41 \( 1 + (-0.918 - 0.394i)T \)
43 \( 1 + (0.440 - 0.897i)T \)
47 \( 1 + (-0.994 + 0.101i)T \)
53 \( 1 + (-0.440 + 0.897i)T \)
59 \( 1 + (0.440 - 0.897i)T \)
61 \( 1 + (0.954 + 0.299i)T \)
67 \( 1 + (0.918 - 0.394i)T \)
71 \( 1 + (-0.979 - 0.201i)T \)
73 \( 1 + (0.528 + 0.848i)T \)
79 \( 1 + (0.918 + 0.394i)T \)
83 \( 1 + (-0.0506 - 0.998i)T \)
89 \( 1 + (-0.440 - 0.897i)T \)
97 \( 1 + (-0.979 - 0.201i)T \)
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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

−24.93708670281040341132986484501, −23.62937620888386227088631649935, −23.040699820031964929673386687, −22.33912040396256575037856008254, −21.10517774670626270592670849740, −20.59186936130466266244939180261, −19.72038841663835414926155955957, −19.261007703828698375276958265600, −17.88594771006873611128199246844, −16.56104561365296152647704233316, −15.86808056104803619545110128715, −14.7811630722604782517007226931, −13.999439172983191786983902963933, −12.94316308828735930630218922871, −11.91793443698367554731775597513, −11.03793418928896755931021345481, −10.05712139114714378275615746282, −9.451376988974951943070086620274, −8.21364139193450686188203547806, −7.069123840496939415689857089069, −4.98208612261417304635930180794, −4.6679605908668995691330144727, −3.472034184861640279694384420276, −2.71110718830337096671519289844, −0.75238193880979355690292850287, 0.52782498080365812244817285555, 2.82071807094832301273231881794, 3.3779639589378150276683694447, 5.13763380812926581795032706400, 6.03558208012880741646152413991, 7.10628011164649377906791757777, 7.98966204232718479322309158432, 8.47185061853202277019846811240, 9.85658303191781345160416184711, 11.617305743725656849922712467878, 12.38406505915313077530117594475, 13.086576236980750822602577493218, 14.16691153700122285457843459885, 15.16341646071326725754462927569, 15.61395894360997479586493324654, 16.82065586791992466000575540157, 17.89801864248956572099405862445, 18.836610770204343444381333784004, 19.18823397947488206485741859149, 20.54873481032781196715539072463, 21.80666411183188146267088236175, 22.68263686028236011847219796648, 23.463590607854487607557641786805, 24.170379144211516141210523195717, 24.96794544091862613686218904331

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