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Properties

Label 1-1339-1339.1042-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.894 - 0.446i$
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  + (0.673 − 0.739i)2-s + (0.779 − 0.626i)3-s + (−0.0922 − 0.995i)4-s + (0.999 − 0.0307i)5-s + (0.0615 − 0.998i)6-s + (0.798 − 0.602i)7-s + (−0.798 − 0.602i)8-s + (0.213 − 0.976i)9-s + (0.650 − 0.759i)10-s + (−0.976 − 0.213i)11-s + (−0.696 − 0.717i)12-s + (0.0922 − 0.995i)14-s + (0.759 − 0.650i)15-s + (−0.982 + 0.183i)16-s + (−0.445 + 0.895i)17-s + (−0.577 − 0.816i)18-s + ⋯
L(s)  = 1  + (0.673 − 0.739i)2-s + (0.779 − 0.626i)3-s + (−0.0922 − 0.995i)4-s + (0.999 − 0.0307i)5-s + (0.0615 − 0.998i)6-s + (0.798 − 0.602i)7-s + (−0.798 − 0.602i)8-s + (0.213 − 0.976i)9-s + (0.650 − 0.759i)10-s + (−0.976 − 0.213i)11-s + (−0.696 − 0.717i)12-s + (0.0922 − 0.995i)14-s + (0.759 − 0.650i)15-s + (−0.982 + 0.183i)16-s + (−0.445 + 0.895i)17-s + (−0.577 − 0.816i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8061721492 - 3.423154315i\)
\(L(\frac12)\) \(\approx\) \(0.8061721492 - 3.423154315i\)
\(L(1)\) \(\approx\) \(1.474456564 - 1.635703850i\)
\(L(1)\) \(\approx\) \(1.474456564 - 1.635703850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.673 - 0.739i)T \)
3 \( 1 + (0.779 - 0.626i)T \)
5 \( 1 + (0.999 - 0.0307i)T \)
7 \( 1 + (0.798 - 0.602i)T \)
11 \( 1 + (-0.976 - 0.213i)T \)
17 \( 1 + (-0.445 + 0.895i)T \)
19 \( 1 + (0.988 + 0.153i)T \)
23 \( 1 + (-0.213 - 0.976i)T \)
29 \( 1 + (-0.850 + 0.526i)T \)
31 \( 1 + (-0.183 - 0.982i)T \)
37 \( 1 + (-0.717 + 0.696i)T \)
41 \( 1 + (-0.526 + 0.850i)T \)
43 \( 1 + (0.969 + 0.243i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.153 - 0.988i)T \)
59 \( 1 + (0.920 - 0.389i)T \)
61 \( 1 + (-0.998 + 0.0615i)T \)
67 \( 1 + (-0.122 + 0.992i)T \)
71 \( 1 + (0.526 - 0.850i)T \)
73 \( 1 + (-0.526 + 0.850i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (0.920 + 0.389i)T \)
89 \( 1 + (0.577 - 0.816i)T \)
97 \( 1 + (-0.0615 + 0.998i)T \)
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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

−21.20324863167739660048167472496, −20.81012581899561857333149873258, −20.160666324352680424488797293215, −18.718894932279136221225567710889, −18.00870272606357073566665340301, −17.453923226041462696245533655639, −16.41280184095252518056313655817, −15.57110857465409829326391997406, −15.29299682782997384948844226277, −14.1359163711708276294108136783, −13.9042188289321232249676221087, −13.16387264788081219496089126760, −12.17578903203126871118772096173, −11.174638198082759256628241410563, −10.25498794223176963922261615945, −9.14053017945668497613906524096, −8.885103727218874170941294361870, −7.650228049975181908606188635363, −7.260124138651694387705962715646, −5.70902411152283669606237945592, −5.30348307225920306296299764794, −4.64228995094099315264448013703, −3.441016842435315278898536551619, −2.56618916541297233626601231580, −1.93357513320640077538504538465, 0.946499939469936834388125255692, 1.810376356345616356740600975787, 2.44239696652566985932141761118, 3.38529301979652700784129226358, 4.37996515559601191491882086417, 5.33179714810315998587653842966, 6.121450903966466257608309540323, 7.06901167905045880652938274892, 8.06017396425951500015102172092, 8.899133318308784293060327712582, 9.84728051273663832629937521540, 10.49930769862676138046815581947, 11.289313139116068433249212809258, 12.34666339740382922893462238342, 13.19674682985914714053146440920, 13.43397647428179237511636226017, 14.36285153401471377580914041018, 14.70858703809051251251488243837, 15.73535844722800140045175372920, 16.97879794751343053242538978151, 17.941757670893728211824527035090, 18.372828313755809330474890534299, 19.103334264638681671868845038060, 20.235833345036680023612733920758, 20.54129774677367425581291140265

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