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Properties

Label 1-3311-3311.1048-r1-0-0
Degree $1$
Conductor $3311$
Sign $0.914 - 0.404i$
Analytic cond. $355.816$
Root an. cond. $355.816$
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  + (0.791 − 0.611i)2-s + (0.337 − 0.941i)3-s + (0.251 − 0.967i)4-s + (−0.575 − 0.817i)5-s + (−0.309 − 0.951i)6-s + (−0.393 − 0.919i)8-s + (−0.772 − 0.635i)9-s + (−0.955 − 0.294i)10-s + (−0.826 − 0.563i)12-s + (−0.858 − 0.512i)13-s + (−0.963 + 0.266i)15-s + (−0.873 − 0.486i)16-s + (−0.0149 − 0.999i)17-s + (−0.999 − 0.0299i)18-s + (−0.887 + 0.460i)19-s + (−0.936 + 0.351i)20-s + ⋯
L(s)  = 1  + (0.791 − 0.611i)2-s + (0.337 − 0.941i)3-s + (0.251 − 0.967i)4-s + (−0.575 − 0.817i)5-s + (−0.309 − 0.951i)6-s + (−0.393 − 0.919i)8-s + (−0.772 − 0.635i)9-s + (−0.955 − 0.294i)10-s + (−0.826 − 0.563i)12-s + (−0.858 − 0.512i)13-s + (−0.963 + 0.266i)15-s + (−0.873 − 0.486i)16-s + (−0.0149 − 0.999i)17-s + (−0.999 − 0.0299i)18-s + (−0.887 + 0.460i)19-s + (−0.936 + 0.351i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2018943639 - 0.04265663955i\)
\(L(\frac12)\) \(\approx\) \(0.2018943639 - 0.04265663955i\)
\(L(1)\) \(\approx\) \(0.6955724144 - 1.066909410i\)
\(L(1)\) \(\approx\) \(0.6955724144 - 1.066909410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.791 - 0.611i)T \)
3 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (-0.575 - 0.817i)T \)
13 \( 1 + (-0.858 - 0.512i)T \)
17 \( 1 + (-0.0149 - 0.999i)T \)
19 \( 1 + (-0.887 + 0.460i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (0.983 + 0.178i)T \)
31 \( 1 + (0.925 + 0.379i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.473 + 0.880i)T \)
47 \( 1 + (-0.887 + 0.460i)T \)
53 \( 1 + (0.575 - 0.817i)T \)
59 \( 1 + (0.599 + 0.800i)T \)
61 \( 1 + (0.925 - 0.379i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (0.936 + 0.351i)T \)
73 \( 1 + (-0.193 - 0.981i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.134 + 0.990i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
97 \( 1 + (-0.936 + 0.351i)T \)
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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

−19.03337882510195168564717445854, −17.61578602860835713486165376062, −17.26449043007159274732804042062, −16.456794498361896081635892286566, −15.66385271202126808008132010710, −15.23999469840610455561730960834, −14.71837540141456383460686558077, −14.11832145572912441762682581451, −13.43741078934443756473815018111, −12.4541354053243950151781859777, −11.72905666684868888700222635638, −11.0750846539448613534889866850, −10.39305351258670030925739903046, −9.54241315099121235127573473009, −8.54682645736278581490404360263, −8.082494209772395309398800301418, −7.15961373685587611064238032888, −6.56783057847072780630427806258, −5.662888908974932108154684810081, −4.82052213250053731802222775390, −4.1161929371150305318338485016, −3.63557179513061591629916282866, −2.71143130154090214754264413378, −2.14583433059681452378274886911, −0.0263053436034420940844326952, 0.767603054212257936859789379517, 1.45233400564822105117877384157, 2.56886282725879216442991391183, 2.97565880279230995161436666813, 4.07213084315999323722289882556, 4.84032109177226580878239107168, 5.412535250125036900620555640920, 6.555028206276298301449019056884, 6.96092953062191458830335479133, 8.109550863354851668355878149197, 8.509451257251041900193080651009, 9.54833948808932977300038332560, 10.22168949771810448552570290904, 11.28655030375901290242931061067, 11.94325527132912648487792799993, 12.35445990634399143084004476321, 13.00794516801287071575102326105, 13.522324216345302115450040254729, 14.45106121335972848154450241746, 14.87402153320813555583599999214, 15.717902389528345472912765885468, 16.51199478037465279900392089913, 17.33652043800885744006709473199, 18.170841154650281039749783417439, 18.93571598267943400099570268836

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