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Properties

Label 1-1339-1339.1097-r0-0-0
Degree $1$
Conductor $1339$
Sign $0.895 - 0.446i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (−0.626 + 0.779i)2-s + (0.982 − 0.183i)3-s + (−0.213 − 0.976i)4-s + (0.717 − 0.696i)5-s + (−0.473 + 0.881i)6-s + (0.0615 − 0.998i)7-s + (0.895 + 0.445i)8-s + (0.932 − 0.361i)9-s + (0.0922 + 0.995i)10-s + (0.988 + 0.153i)11-s + (−0.389 − 0.920i)12-s + (0.739 + 0.673i)14-s + (0.577 − 0.816i)15-s + (−0.908 + 0.417i)16-s + (−0.881 + 0.473i)17-s + (−0.303 + 0.952i)18-s + ⋯
L(s)  = 1  + (−0.626 + 0.779i)2-s + (0.982 − 0.183i)3-s + (−0.213 − 0.976i)4-s + (0.717 − 0.696i)5-s + (−0.473 + 0.881i)6-s + (0.0615 − 0.998i)7-s + (0.895 + 0.445i)8-s + (0.932 − 0.361i)9-s + (0.0922 + 0.995i)10-s + (0.988 + 0.153i)11-s + (−0.389 − 0.920i)12-s + (0.739 + 0.673i)14-s + (0.577 − 0.816i)15-s + (−0.908 + 0.417i)16-s + (−0.881 + 0.473i)17-s + (−0.303 + 0.952i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $0.895 - 0.446i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ 0.895 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.028390373 - 0.4774617268i\)
\(L(\frac12)\) \(\approx\) \(2.028390373 - 0.4774617268i\)
\(L(1)\) \(\approx\) \(1.349469930 - 0.04126158646i\)
\(L(1)\) \(\approx\) \(1.349469930 - 0.04126158646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.626 + 0.779i)T \)
3 \( 1 + (0.982 - 0.183i)T \)
5 \( 1 + (0.717 - 0.696i)T \)
7 \( 1 + (0.0615 - 0.998i)T \)
11 \( 1 + (0.988 + 0.153i)T \)
17 \( 1 + (-0.881 + 0.473i)T \)
19 \( 1 + (0.943 - 0.332i)T \)
23 \( 1 + (-0.932 - 0.361i)T \)
29 \( 1 + (0.969 - 0.243i)T \)
31 \( 1 + (0.995 + 0.0922i)T \)
37 \( 1 + (0.798 + 0.602i)T \)
41 \( 1 + (0.717 + 0.696i)T \)
43 \( 1 + (-0.389 + 0.920i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.650 + 0.759i)T \)
59 \( 1 + (-0.0615 - 0.998i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (-0.833 + 0.552i)T \)
71 \( 1 + (-0.243 + 0.969i)T \)
73 \( 1 + (-0.961 + 0.273i)T \)
79 \( 1 + (-0.273 + 0.961i)T \)
83 \( 1 + (-0.833 - 0.552i)T \)
89 \( 1 + (0.673 - 0.739i)T \)
97 \( 1 + (0.473 - 0.881i)T \)
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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

−20.934636825681326366775644254735, −20.10601009421470810501279744973, −19.44762821913131292180604189755, −18.79356504511970756894492832936, −18.03917255897954732727910700364, −17.630750432308538138278611553422, −16.36517368828874610002949505945, −15.660437739106866105901970095214, −14.740499805720388261625006283678, −13.85882644183659454738155508472, −13.51919737887570975238919284626, −12.26083924348624650401499181276, −11.72123654160031452096605749731, −10.68370394853872700249437837044, −9.907168493499253294296207182503, −9.22393947489615087436703742119, −8.81105252730758309357197849689, −7.79496422824505178248748536661, −6.94385915877471721537352059481, −5.95496859759840237996086198309, −4.630502630227572082839096446945, −3.63763721995355979330585549279, −2.80924693804560259273487466704, −2.21286952001007077336468085914, −1.3726501719823198481734180677, 1.025029228679342167948990516874, 1.51840485735087838485028840719, 2.72974585841433730613608420258, 4.33560773063093399824218628771, 4.53858496422677340941979890807, 6.14834884956721913330885129749, 6.60868334608392124758142510385, 7.61416271542632113936184769498, 8.29440583579585706967997342713, 9.040822593371499011469728832157, 9.75627485246790874727713790979, 10.22458469675949480143025338, 11.465683443188693900597924729028, 12.70446691991730384937958285658, 13.55523368332861748224373313426, 14.01795264366208580239929650651, 14.583683041973021460115285320520, 15.6822278457633406999224423649, 16.24521831245537609797554042959, 17.26180495601708990122171980516, 17.5920597598262317181743028544, 18.450410100698048199189509819622, 19.534919228150244894858219772485, 20.02837564927184125086616734258, 20.38469218544349940375698328173

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