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Properties

Label 1-311-311.10-r0-0-0
Degree $1$
Conductor $311$
Sign $0.681 - 0.731i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.983 + 0.181i)2-s + (−0.476 + 0.879i)3-s + (0.934 − 0.356i)4-s + (0.494 − 0.869i)5-s + (0.309 − 0.951i)6-s + (−0.954 + 0.299i)7-s + (−0.853 + 0.520i)8-s + (−0.546 − 0.837i)9-s + (−0.328 + 0.944i)10-s + (0.918 − 0.394i)11-s + (−0.131 + 0.991i)12-s + (−0.0506 + 0.998i)13-s + (0.884 − 0.467i)14-s + (0.528 + 0.848i)15-s + (0.745 − 0.666i)16-s + (−0.289 − 0.957i)17-s + ⋯
L(s)  = 1  + (−0.983 + 0.181i)2-s + (−0.476 + 0.879i)3-s + (0.934 − 0.356i)4-s + (0.494 − 0.869i)5-s + (0.309 − 0.951i)6-s + (−0.954 + 0.299i)7-s + (−0.853 + 0.520i)8-s + (−0.546 − 0.837i)9-s + (−0.328 + 0.944i)10-s + (0.918 − 0.394i)11-s + (−0.131 + 0.991i)12-s + (−0.0506 + 0.998i)13-s + (0.884 − 0.467i)14-s + (0.528 + 0.848i)15-s + (0.745 − 0.666i)16-s + (−0.289 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4820390446 - 0.2096292220i\)
\(L(\frac12)\) \(\approx\) \(0.4820390446 - 0.2096292220i\)
\(L(1)\) \(\approx\) \(0.5682778716 + 0.01805373967i\)
\(L(1)\) \(\approx\) \(0.5682778716 + 0.01805373967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.983 + 0.181i)T \)
3 \( 1 + (-0.476 + 0.879i)T \)
5 \( 1 + (0.494 - 0.869i)T \)
7 \( 1 + (-0.954 + 0.299i)T \)
11 \( 1 + (0.918 - 0.394i)T \)
13 \( 1 + (-0.0506 + 0.998i)T \)
17 \( 1 + (-0.289 - 0.957i)T \)
19 \( 1 + (-0.910 - 0.412i)T \)
23 \( 1 + (0.996 + 0.0809i)T \)
29 \( 1 + (-0.989 + 0.141i)T \)
31 \( 1 + (-0.328 - 0.944i)T \)
37 \( 1 + (-0.0101 - 0.999i)T \)
41 \( 1 + (-0.250 + 0.968i)T \)
43 \( 1 + (0.595 - 0.803i)T \)
47 \( 1 + (-0.440 - 0.897i)T \)
53 \( 1 + (-0.579 - 0.814i)T \)
59 \( 1 + (0.947 + 0.318i)T \)
61 \( 1 + (0.979 - 0.201i)T \)
67 \( 1 + (0.843 - 0.537i)T \)
71 \( 1 + (0.960 + 0.279i)T \)
73 \( 1 + (-0.703 - 0.710i)T \)
79 \( 1 + (0.771 - 0.635i)T \)
83 \( 1 + (0.528 - 0.848i)T \)
89 \( 1 + (-0.954 - 0.299i)T \)
97 \( 1 + (-0.941 + 0.337i)T \)
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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

−25.41916612558196643947415666555, −24.88715984813294429607311829049, −23.58610076753582037851992640679, −22.56978779287704721767513808422, −22.01879314531390385995248459258, −20.61262647972890342882797697871, −19.41152624306551872813573218614, −19.18826510184118641524498531177, −18.119473828200868889627342358006, −17.25289992010311249182690691322, −16.891111886163345787538448897524, −15.45062631886449573724781444451, −14.46275813216503639416425910224, −12.99934097650291924943879092918, −12.52988872350587295300910371812, −11.15799076881222826347602517459, −10.53696161288976988480915106219, −9.60244948481080239595781036340, −8.37330762705357958091391934699, −7.17203709775030229283533219928, −6.59956008314439471381977082704, −5.8315737343049918783998614512, −3.54885294580226935111116685776, −2.440447134123725047854877194181, −1.296197537344909580348088434547, 0.52505583010557145213793674074, 2.19962531026402430806384958822, 3.74333118510285125262870708375, 5.11125512874874649926860647466, 6.14306513798594446727797728911, 6.88492121467316281181678167910, 8.80292179509257817143287641226, 9.19382522605548609258294293523, 9.80435325925398590899467431655, 11.12745493211612932123253763894, 11.81954361821727652502696494443, 13.0177461959004149884138212827, 14.45080018922293713875576227346, 15.48284144031490650736784242384, 16.55667935867541162935509171119, 16.62640479448769249072178318185, 17.6149252793657618711350384838, 18.83237203975791113788983947897, 19.695777631752840816998654831610, 20.61910575893555710421542789944, 21.43431627099321999005802463754, 22.25807154096234965864911187619, 23.4677503600286224018382841721, 24.46602475254339819755016626, 25.3085493800001541264985648614

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