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Properties

Label 1-3311-3311.1417-r1-0-0
Degree $1$
Conductor $3311$
Sign $0.843 + 0.537i$
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  + (0.251 − 0.967i)2-s + (0.772 + 0.635i)3-s + (−0.873 − 0.486i)4-s + (0.337 − 0.941i)5-s + (0.809 − 0.587i)6-s + (−0.691 + 0.722i)8-s + (0.193 + 0.981i)9-s + (−0.826 − 0.563i)10-s + (−0.365 − 0.930i)12-s + (−0.473 − 0.880i)13-s + (0.858 − 0.512i)15-s + (0.525 + 0.850i)16-s + (0.999 − 0.0299i)17-s + (0.998 + 0.0598i)18-s + (−0.575 + 0.817i)19-s + (−0.753 + 0.657i)20-s + ⋯
L(s)  = 1  + (0.251 − 0.967i)2-s + (0.772 + 0.635i)3-s + (−0.873 − 0.486i)4-s + (0.337 − 0.941i)5-s + (0.809 − 0.587i)6-s + (−0.691 + 0.722i)8-s + (0.193 + 0.981i)9-s + (−0.826 − 0.563i)10-s + (−0.365 − 0.930i)12-s + (−0.473 − 0.880i)13-s + (0.858 − 0.512i)15-s + (0.525 + 0.850i)16-s + (0.999 − 0.0299i)17-s + (0.998 + 0.0598i)18-s + (−0.575 + 0.817i)19-s + (−0.753 + 0.657i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ 0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594452373 + 0.4650521613i\)
\(L(\frac12)\) \(\approx\) \(1.594452373 + 0.4650521613i\)
\(L(1)\) \(\approx\) \(1.191181291 - 0.5129496432i\)
\(L(1)\) \(\approx\) \(1.191181291 - 0.5129496432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.251 - 0.967i)T \)
3 \( 1 + (0.772 + 0.635i)T \)
5 \( 1 + (0.337 - 0.941i)T \)
13 \( 1 + (-0.473 - 0.880i)T \)
17 \( 1 + (0.999 - 0.0299i)T \)
19 \( 1 + (-0.575 + 0.817i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (-0.712 - 0.701i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (0.550 + 0.834i)T \)
47 \( 1 + (-0.575 + 0.817i)T \)
53 \( 1 + (-0.337 - 0.941i)T \)
59 \( 1 + (0.280 - 0.959i)T \)
61 \( 1 + (-0.712 + 0.701i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (0.925 - 0.379i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (-0.753 + 0.657i)T \)
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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

−18.52254402755988366669494011334, −17.90509548719107138257940887272, −17.371965319635421232880798021186, −16.55264767155735613039074859027, −15.59810719655520950563643107713, −15.05335436898469209944078050646, −14.40538215339706543187448867398, −13.81569230165615662624789009311, −13.56122167283422847254898803462, −12.32847070736497139546863534282, −12.076895476111038962959902482, −10.81941096961529608426129717941, −9.829322465900139517609498082885, −9.31879957064946310592093015080, −8.52101657493049891317680668659, −7.71237678316065756100430906610, −7.136858943659037231677071531017, −6.59920210512252227520459216961, −5.924382449813862878544474970137, −4.96392260615163841207371429507, −3.91735944222722709332227395267, −3.2960974419748920931971098499, −2.45478159335700647395775434122, −1.56928375375129838738316846210, −0.220811233789042704464875351811, 0.8712629731722240379122058077, 1.795172686971596265471747176588, 2.497275718523862162819078623123, 3.38762861497733125655899611854, 4.04565969440609519046764232167, 4.8697471943657045491779910870, 5.36401115774154588222364520451, 6.22311702103345459687942816303, 7.93290836690090161005713232915, 8.13374070874752243895330980592, 9.049129046486616628259791933732, 9.70674111788723613105938967928, 10.209365729881900343452087253611, 10.78838410197392793422089197897, 11.94524056175605995538914486792, 12.56005243915404524720275705476, 13.022753964303497109694127597846, 13.89969139982336458238438679377, 14.46542331653009008532634580784, 14.99932667706192200119165038812, 16.05388419043969918911936738326, 16.58772347656235234240894136070, 17.45670558926073320812368329542, 18.168665941624318339459815330833, 19.10326527823085026089356281860

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