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Properties

Label 1-1337-1337.1084-r1-0-0
Degree $1$
Conductor $1337$
Sign $0.301 - 0.953i$
Analytic cond. $143.680$
Root an. cond. $143.680$
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  + (−0.213 − 0.976i)2-s + (0.277 + 0.960i)3-s + (−0.909 + 0.416i)4-s + (0.677 + 0.735i)5-s + (0.879 − 0.475i)6-s + (0.601 + 0.799i)8-s + (−0.846 + 0.533i)9-s + (0.574 − 0.818i)10-s + (−0.986 + 0.164i)11-s + (−0.652 − 0.757i)12-s + (−0.828 − 0.560i)13-s + (−0.518 + 0.854i)15-s + (0.652 − 0.757i)16-s + (−0.965 + 0.261i)17-s + (0.701 + 0.712i)18-s + (0.574 + 0.818i)19-s + ⋯
L(s)  = 1  + (−0.213 − 0.976i)2-s + (0.277 + 0.960i)3-s + (−0.909 + 0.416i)4-s + (0.677 + 0.735i)5-s + (0.879 − 0.475i)6-s + (0.601 + 0.799i)8-s + (−0.846 + 0.533i)9-s + (0.574 − 0.818i)10-s + (−0.986 + 0.164i)11-s + (−0.652 − 0.757i)12-s + (−0.828 − 0.560i)13-s + (−0.518 + 0.854i)15-s + (0.652 − 0.757i)16-s + (−0.965 + 0.261i)17-s + (0.701 + 0.712i)18-s + (0.574 + 0.818i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $0.301 - 0.953i$
Analytic conductor: \(143.680\)
Root analytic conductor: \(143.680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (1:\ ),\ 0.301 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5204820366 - 0.3811099115i\)
\(L(\frac12)\) \(\approx\) \(0.5204820366 - 0.3811099115i\)
\(L(1)\) \(\approx\) \(0.7935191464 + 0.03643131074i\)
\(L(1)\) \(\approx\) \(0.7935191464 + 0.03643131074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.213 - 0.976i)T \)
3 \( 1 + (0.277 + 0.960i)T \)
5 \( 1 + (0.677 + 0.735i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (-0.828 - 0.560i)T \)
17 \( 1 + (-0.965 + 0.261i)T \)
19 \( 1 + (0.574 + 0.818i)T \)
23 \( 1 + (-0.956 + 0.293i)T \)
29 \( 1 + (-0.973 - 0.229i)T \)
31 \( 1 + (-0.245 - 0.969i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (0.401 + 0.915i)T \)
43 \( 1 + (0.180 + 0.983i)T \)
47 \( 1 + (0.148 - 0.988i)T \)
53 \( 1 + (0.701 - 0.712i)T \)
59 \( 1 + (0.768 + 0.639i)T \)
61 \( 1 + (0.934 - 0.355i)T \)
67 \( 1 + (-0.627 - 0.778i)T \)
71 \( 1 + (-0.995 + 0.0990i)T \)
73 \( 1 + (0.461 - 0.887i)T \)
79 \( 1 + (-0.973 + 0.229i)T \)
83 \( 1 + (0.956 + 0.293i)T \)
89 \( 1 + (0.724 - 0.689i)T \)
97 \( 1 + (-0.997 + 0.0660i)T \)
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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

−20.6622638291670395953377256375, −20.02892238840917910575230837817, −19.16179108721128467331712853605, −18.38680692614004757063601241638, −17.72717604888953756276639033496, −17.28549907804543346025262773236, −16.31948114389051882655559999918, −15.690131263809672885746801126571, −14.631399449511136049312502098618, −13.89732305832850869145369787268, −13.36184771171658377909091023618, −12.75009794358197626251934378704, −11.860761860832374443874771183266, −10.59408603737412806373753464800, −9.56586321292343445495615000292, −8.94418889455066752218654080695, −8.28586959910597822545517727698, −7.34221118791364760521961219870, −6.80838089477610839823428373880, −5.763314918614513423678049979844, −5.19642898071961343401987661096, −4.2451935074388913895157340061, −2.67743143898862062971425737331, −1.829067688135103697684599343786, −0.67733311133402452794228950441, 0.17519087982810257526935542767, 2.033788606162469451325232875957, 2.469646803080719764898920757399, 3.403763870236371489616919051703, 4.21430865469298746302637795032, 5.262341356999250045313783748833, 5.85326717894605134399842515842, 7.50624814240225879961527641050, 8.08309328203751696165745344544, 9.26342144495532215989491054359, 9.82762962025784746040010358304, 10.35434159900198889713310337825, 11.02701113022123502587473072228, 11.80773887132218625790013327857, 13.04634202883486326085310586598, 13.478538841906769333645659762536, 14.55377387100491607933225263608, 14.93628317875447770403614673279, 16.05504922489938984658360669007, 16.9061273438072343983431531799, 17.8243004918826874373192914833, 18.18238464017587895818032674732, 19.22557324990342001687991693098, 19.9763481606864147331654691937, 20.60637866539747463890152760923

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