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Properties

Label 1-1339-1339.460-r0-0-0
Degree $1$
Conductor $1339$
Sign $0.983 - 0.181i$
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  + (−0.999 + 0.0307i)2-s + (0.273 + 0.961i)3-s + (0.998 − 0.0615i)4-s + (0.626 − 0.779i)5-s + (−0.303 − 0.952i)6-s + (−0.417 − 0.908i)7-s + (−0.995 + 0.0922i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (0.473 − 0.881i)11-s + (0.332 + 0.943i)12-s + (0.445 + 0.895i)14-s + (0.920 + 0.389i)15-s + (0.992 − 0.122i)16-s + (0.952 + 0.303i)17-s + (0.833 − 0.552i)18-s + ⋯
L(s)  = 1  + (−0.999 + 0.0307i)2-s + (0.273 + 0.961i)3-s + (0.998 − 0.0615i)4-s + (0.626 − 0.779i)5-s + (−0.303 − 0.952i)6-s + (−0.417 − 0.908i)7-s + (−0.995 + 0.0922i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (0.473 − 0.881i)11-s + (0.332 + 0.943i)12-s + (0.445 + 0.895i)14-s + (0.920 + 0.389i)15-s + (0.992 − 0.122i)16-s + (0.952 + 0.303i)17-s + (0.833 − 0.552i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167503861 - 0.1065554683i\)
\(L(\frac12)\) \(\approx\) \(1.167503861 - 0.1065554683i\)
\(L(1)\) \(\approx\) \(0.8575981557 + 0.03757561885i\)
\(L(1)\) \(\approx\) \(0.8575981557 + 0.03757561885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.999 + 0.0307i)T \)
3 \( 1 + (0.273 + 0.961i)T \)
5 \( 1 + (0.626 - 0.779i)T \)
7 \( 1 + (-0.417 - 0.908i)T \)
11 \( 1 + (0.473 - 0.881i)T \)
17 \( 1 + (0.952 + 0.303i)T \)
19 \( 1 + (-0.717 + 0.696i)T \)
23 \( 1 + (0.850 + 0.526i)T \)
29 \( 1 + (-0.153 + 0.988i)T \)
31 \( 1 + (0.798 - 0.602i)T \)
37 \( 1 + (-0.183 + 0.982i)T \)
41 \( 1 + (0.626 + 0.779i)T \)
43 \( 1 + (0.332 - 0.943i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.969 + 0.243i)T \)
59 \( 1 + (0.417 - 0.908i)T \)
61 \( 1 + (0.739 - 0.673i)T \)
67 \( 1 + (0.577 + 0.816i)T \)
71 \( 1 + (0.988 - 0.153i)T \)
73 \( 1 + (0.361 - 0.932i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (0.577 - 0.816i)T \)
89 \( 1 + (0.895 - 0.445i)T \)
97 \( 1 + (0.303 + 0.952i)T \)
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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

−20.95740660406599751891009786284, −19.74828951191272270022470315226, −19.1853969141433710291056326128, −18.75148272144333285368135839479, −17.86740217025870611240476506600, −17.54573732820542313344721767092, −16.67189459184446279106041647016, −15.49723336928182329448520804313, −14.846222499305038967572751820285, −14.24201327941946186136746854069, −13.01634911757007126674327452257, −12.39505838213382115319907975906, −11.647770574331369695547697873208, −10.80705429496709263193618765846, −9.73318421709438031150843808667, −9.28562324729362728012135548561, −8.41813523942195502077945950903, −7.48574329687478160398596309794, −6.71399524662761912065622217231, −6.30738360968282176573242324528, −5.32678769878202008599445700316, −3.45133295324543975946975691400, −2.47402309859488840046348424094, −2.19434207608637295332576149711, −0.96781718744338252177377456268, 0.75796901525504459570587100357, 1.70615872139316304561411776617, 3.06860634703439150879856501307, 3.71445772116110379023363665413, 4.906114005391272073534775833060, 5.87320826934381879873199564763, 6.56668537373490833137521525660, 7.91442193756375670418064268237, 8.4226619350110111005533471774, 9.36412314268979288845529820611, 9.784862331417974599868426100065, 10.569541000913370013813753682630, 11.20734746379644541423918628239, 12.291188420741563408337937230739, 13.26677797918402223569690338853, 14.177236065956863015393432751238, 14.83478332653525925325814130104, 16.00238504818362594906889885694, 16.42267650017304606427352371074, 17.10077540196029945641191241026, 17.36802671195035960924072986940, 18.89931344022309548851242038260, 19.320168123361183362848026322051, 20.23145059066797134470721435624, 20.7516387593793586541466736771

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