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Properties

Label 1-311-311.144-r0-0-0
Degree $1$
Conductor $311$
Sign $0.856 + 0.516i$
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  + (−0.989 − 0.141i)2-s + (0.884 + 0.467i)3-s + (0.960 + 0.279i)4-s + (0.992 + 0.121i)5-s + (−0.809 − 0.587i)6-s + (0.688 − 0.724i)7-s + (−0.910 − 0.412i)8-s + (0.562 + 0.826i)9-s + (−0.965 − 0.260i)10-s + (0.528 + 0.848i)11-s + (0.717 + 0.696i)12-s + (−0.612 + 0.790i)13-s + (−0.784 + 0.620i)14-s + (0.820 + 0.571i)15-s + (0.843 + 0.537i)16-s + (−0.546 + 0.837i)17-s + ⋯
L(s)  = 1  + (−0.989 − 0.141i)2-s + (0.884 + 0.467i)3-s + (0.960 + 0.279i)4-s + (0.992 + 0.121i)5-s + (−0.809 − 0.587i)6-s + (0.688 − 0.724i)7-s + (−0.910 − 0.412i)8-s + (0.562 + 0.826i)9-s + (−0.965 − 0.260i)10-s + (0.528 + 0.848i)11-s + (0.717 + 0.696i)12-s + (−0.612 + 0.790i)13-s + (−0.784 + 0.620i)14-s + (0.820 + 0.571i)15-s + (0.843 + 0.537i)16-s + (−0.546 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.856 + 0.516i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.856 + 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352931122 + 0.3768219876i\)
\(L(\frac12)\) \(\approx\) \(1.352931122 + 0.3768219876i\)
\(L(1)\) \(\approx\) \(1.139848552 + 0.1714917011i\)
\(L(1)\) \(\approx\) \(1.139848552 + 0.1714917011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.989 - 0.141i)T \)
3 \( 1 + (0.884 + 0.467i)T \)
5 \( 1 + (0.992 + 0.121i)T \)
7 \( 1 + (0.688 - 0.724i)T \)
11 \( 1 + (0.528 + 0.848i)T \)
13 \( 1 + (-0.612 + 0.790i)T \)
17 \( 1 + (-0.546 + 0.837i)T \)
19 \( 1 + (0.191 - 0.981i)T \)
23 \( 1 + (0.111 - 0.993i)T \)
29 \( 1 + (-0.832 + 0.554i)T \)
31 \( 1 + (-0.965 + 0.260i)T \)
37 \( 1 + (-0.983 - 0.181i)T \)
41 \( 1 + (0.151 - 0.988i)T \)
43 \( 1 + (-0.476 - 0.879i)T \)
47 \( 1 + (0.347 - 0.937i)T \)
53 \( 1 + (-0.131 + 0.991i)T \)
59 \( 1 + (0.902 + 0.431i)T \)
61 \( 1 + (-0.874 - 0.485i)T \)
67 \( 1 + (-0.703 - 0.710i)T \)
71 \( 1 + (0.385 + 0.922i)T \)
73 \( 1 + (-0.0910 - 0.995i)T \)
79 \( 1 + (0.986 - 0.161i)T \)
83 \( 1 + (0.820 - 0.571i)T \)
89 \( 1 + (0.688 + 0.724i)T \)
97 \( 1 + (0.996 - 0.0809i)T \)
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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

−25.02924853483631266747050260029, −24.740665999091428026857385298161, −24.04278757812124916910526197523, −22.25310054010736698335815261779, −21.195971437949404549085248283192, −20.60812268033949899958926689469, −19.647609864238404153150643544590, −18.710472711293214963317326073948, −18.03135909225506265712568626875, −17.34353556036637935748928063104, −16.183009817246303247669009635065, −15.005032606033650194510879873846, −14.393738490770035247037426390368, −13.32680747516046528509553146050, −12.14850895947354710850527549986, −11.151749596659254918820058331403, −9.76359525100558254659243821420, −9.21131099872369729175999033720, −8.30553693967027341546899681111, −7.4673068455778367876493686129, −6.226570488727010811201758160039, −5.34955643000257171589748224921, −3.19295893986269861426209934307, −2.17938348624821861671434447127, −1.30612719478942793533320757934, 1.71639678688970585239532616663, 2.22447308486293926170496359180, 3.78873671686864666283954346641, 4.982529869383015669486452008713, 6.79523041507419937884869322851, 7.35730344385239008663880686213, 8.80865292579507678705821527218, 9.234372503788397885794179998060, 10.348659184776355891152216462316, 10.85003800141660987788331902114, 12.347497454228758791153704134539, 13.554779737743829324087393647648, 14.55213241511655269429404746069, 15.14927967057102393208625757301, 16.588748414568082139554723910729, 17.19499984059868669805355354680, 18.02525619237116831767385525105, 19.10476611290124474722625109121, 20.10513679178090690261922596046, 20.52528496223317227868358038523, 21.54655171851212478607413745633, 22.14593240484995030759422355506, 24.071808094244419271773342363, 24.62389890259167667755630569448, 25.67418981689261897903571611579

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