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Properties

Label 1-1309-1309.208-r0-0-0
Degree $1$
Conductor $1309$
Sign $0.510 - 0.859i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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  + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)12-s + 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + i·20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)12-s + 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ 0.510 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9347876938 - 0.5319010269i\)
\(L(\frac12)\) \(\approx\) \(0.9347876938 - 0.5319010269i\)
\(L(1)\) \(\approx\) \(0.7112388478 - 0.3675296380i\)
\(L(1)\) \(\approx\) \(0.7112388478 - 0.3675296380i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + iT \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−21.25257709528418099712351147003, −20.44338435580411513011649143365, −19.22903609807311591971967750581, −18.47249187760450279647998856470, −17.84725635685901681438707695191, −17.30891239350935079984010431067, −16.663075581321437561459959803219, −15.64625522346125472235748507500, −15.38810593180582239951808984307, −14.267688970062885299571411710510, −13.61427795730721914539947239934, −12.77206851876492477709426550613, −11.39575801409504198180103053889, −10.81184405755048996374811372689, −10.15639900795398901748104232457, −9.26592561806805875389630322001, −8.80394427411402720545643209677, −7.28800932436616720066149917753, −6.83660408372511920933352621134, −5.773123447634528465872338491810, −5.535860975901879093648292077500, −4.44947379623231860300266837650, −3.35871806695801112228167937241, −1.83549922896055039185848245317, −0.75345588571394919943708651855, 0.99886280233091995514585786470, 1.49989496550486233906441197022, 2.53117209755856052349324866977, 3.73663419265483184000839340820, 4.82508107391670071041846786844, 5.57679665478579045295665454927, 6.49024359887477994131185557146, 7.46865141670357395005076466076, 8.432519407646299744082818175738, 9.16311357379235081544682354452, 10.098416965271157118083268839807, 10.74960786908833851698083299966, 11.44545661294220519146983140126, 12.38947641965207507919721703373, 12.89444222790091235428003628033, 13.54663184490888188586856464574, 14.364950634095525377345361271474, 15.993761564643440324145495149178, 16.56229579861976001767676224852, 17.1700287643382091222940144856, 17.99644883161940365765525418168, 18.4167799209493967314131057167, 19.116835880771351018652933097234, 20.19909542787605748155662894276, 20.838480395649596402918330612425

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