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Properties

Label 1-1339-1339.1087-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.687 - 0.726i$
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  + (−0.866 − 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + 9-s + 10-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + 9-s + 10-s + (0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.687 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03733954561 - 0.08669309771i\)
\(L(\frac12)\) \(\approx\) \(0.03733954561 - 0.08669309771i\)
\(L(1)\) \(\approx\) \(0.3728209315 + 0.01129357614i\)
\(L(1)\) \(\approx\) \(0.3728209315 + 0.01129357614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + iT \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−20.94631312286695538832350620069, −20.224935202843453116610991750317, −19.42949269187764425842558987427, −18.96286349477115800973945883315, −18.02025458107357390846838794053, −17.21889521518849649011285967423, −16.57980173554660518043100940918, −16.196752532529663087200440261458, −15.4653573743309402949236866535, −14.59375153424422652417843603339, −13.48389158923638134149563910114, −12.32923126594868937260848157630, −11.944746987341029200494701746241, −11.04532301517547049220436462395, −10.156581077177012768284704820467, −9.610825838752566538229906546958, −8.6662382996114688752783499086, −7.61102984695261091571664066193, −7.02509450678586852015885425919, −6.31757615348740944655378751681, −5.42287705442683108847090294543, −4.42154256764575918788218284421, −3.63750294586048118592702633727, −1.898806834944076803631329449667, −0.82333531964692505713192013960, 0.08375851756864013461534625034, 1.32241322440704873023045415042, 2.556381186444890778521894311436, 3.67832937382665197744544207488, 4.128549967330533114614164774343, 5.76203939913787513108004067633, 6.52401455794369730921276328714, 7.07121355285862592162644671341, 8.15878364424552614965594841962, 8.91290827492431220534382857332, 9.96394701077997423398546570894, 10.506444064450891879192095546383, 11.40308858780694007908419083135, 11.88930255908589361073984546239, 12.531135112059636525296712015859, 13.335063107382747355144019948989, 14.89451356611982519177909310843, 15.49393314102788666942270700821, 16.42468665144147976383926732052, 16.73645383202440783458033368463, 17.64750183561225472434757396916, 18.59718334168730589883811428485, 18.97440073067782216530592381184, 19.57709317717273743452206685333, 20.41589357777130924252460560049

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