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Properties

Label 1-3311-3311.1158-r1-0-0
Degree $1$
Conductor $3311$
Sign $-0.622 - 0.782i$
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  + (−0.163 + 0.986i)2-s + (0.842 + 0.538i)3-s + (−0.946 − 0.323i)4-s + (0.691 + 0.722i)5-s + (−0.669 + 0.743i)6-s + (0.473 − 0.880i)8-s + (0.420 + 0.907i)9-s + (−0.826 + 0.563i)10-s + (−0.623 − 0.781i)12-s + (0.925 + 0.379i)13-s + (0.193 + 0.981i)15-s + (0.791 + 0.611i)16-s + (−0.134 + 0.990i)17-s + (−0.963 + 0.266i)18-s + (0.393 + 0.919i)19-s + (−0.420 − 0.907i)20-s + ⋯
L(s)  = 1  + (−0.163 + 0.986i)2-s + (0.842 + 0.538i)3-s + (−0.946 − 0.323i)4-s + (0.691 + 0.722i)5-s + (−0.669 + 0.743i)6-s + (0.473 − 0.880i)8-s + (0.420 + 0.907i)9-s + (−0.826 + 0.563i)10-s + (−0.623 − 0.781i)12-s + (0.925 + 0.379i)13-s + (0.193 + 0.981i)15-s + (0.791 + 0.611i)16-s + (−0.134 + 0.990i)17-s + (−0.963 + 0.266i)18-s + (0.393 + 0.919i)19-s + (−0.420 − 0.907i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ -0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.201959299 + 2.491917039i\)
\(L(\frac12)\) \(\approx\) \(-1.201959299 + 2.491917039i\)
\(L(1)\) \(\approx\) \(0.8133230038 + 1.127148583i\)
\(L(1)\) \(\approx\) \(0.8133230038 + 1.127148583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.163 + 0.986i)T \)
3 \( 1 + (0.842 + 0.538i)T \)
5 \( 1 + (0.691 + 0.722i)T \)
13 \( 1 + (0.925 + 0.379i)T \)
17 \( 1 + (-0.134 + 0.990i)T \)
19 \( 1 + (0.393 + 0.919i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.887 + 0.460i)T \)
31 \( 1 + (0.163 - 0.986i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.963 + 0.266i)T \)
47 \( 1 + (-0.992 - 0.119i)T \)
53 \( 1 + (-0.280 - 0.959i)T \)
59 \( 1 + (-0.525 + 0.850i)T \)
61 \( 1 + (0.163 + 0.986i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (0.420 - 0.907i)T \)
73 \( 1 + (-0.983 - 0.178i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.163 + 0.986i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (0.995 + 0.0896i)T \)
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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

−18.30368216419719379507992531116, −17.64366955499930177927705052200, −17.40903153105966907778316734503, −16.02535580500044721403909475752, −15.62302953893242077037178577394, −14.24678617246316825345770491361, −13.887503792309610932473822461673, −13.298553885885535014460150019, −12.80222268074815593802905527167, −11.99125764020409288154203487406, −11.3737996475425934706748619445, −10.33435256604926161716775048261, −9.64761427932381459173175635521, −9.065350974272704730740155602145, −8.53864473899185306891590310710, −7.82580165945762831948942082936, −6.89903706765352627680008700295, −5.884442714853055606632156177647, −4.9957443709252501399487227938, −4.271407403095555044099195084873, −3.18169623683538807684674006720, −2.77413134277346276058604943691, −1.713257746562098329952329080923, −1.18796703889474025716962997551, −0.38573214157743362913202209789, 1.270657847838627374495942517076, 2.06627374095668604450787643088, 3.18023772446149849063478961588, 3.89125409060540266259219138507, 4.594113407082959355778172321208, 5.670675315632015176297869092984, 6.21326744462123164346706540463, 6.962373408071430699392926802806, 7.85988072350302419675562567487, 8.462433952564213689824198559891, 9.10301552511299271591980743985, 9.89253386739739102841525192963, 10.403258840753624434595510008054, 11.03192983171584934586604275962, 12.441218932239299634713736232238, 13.29915637961534456481593373978, 13.79814178703123621725637607369, 14.49866417815989064045978796010, 14.81236109011455495559923886411, 15.64655147998557369410241008478, 16.333185202748040344553088866737, 16.828653474022037956610447806081, 17.89450106155257166006525147949, 18.26455499343626383123834202414, 19.09973306817114992487895795664

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