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Properties

Label 1-1049-1049.1040-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.587 + 0.808i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (0.649 + 0.760i)2-s + (−0.999 − 0.0119i)3-s + (−0.155 + 0.987i)4-s + (0.385 + 0.922i)5-s + (−0.640 − 0.767i)6-s + (0.996 − 0.0838i)7-s + (−0.851 + 0.524i)8-s + (0.999 + 0.0239i)9-s + (−0.450 + 0.892i)10-s + (0.295 − 0.955i)11-s + (0.167 − 0.985i)12-s + (−0.107 + 0.994i)13-s + (0.711 + 0.702i)14-s + (−0.374 − 0.927i)15-s + (−0.951 − 0.306i)16-s + (0.940 − 0.340i)17-s + ⋯
L(s)  = 1  + (0.649 + 0.760i)2-s + (−0.999 − 0.0119i)3-s + (−0.155 + 0.987i)4-s + (0.385 + 0.922i)5-s + (−0.640 − 0.767i)6-s + (0.996 − 0.0838i)7-s + (−0.851 + 0.524i)8-s + (0.999 + 0.0239i)9-s + (−0.450 + 0.892i)10-s + (0.295 − 0.955i)11-s + (0.167 − 0.985i)12-s + (−0.107 + 0.994i)13-s + (0.711 + 0.702i)14-s + (−0.374 − 0.927i)15-s + (−0.951 − 0.306i)16-s + (0.940 − 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.587 + 0.808i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.587 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8794034044 + 1.726450708i\)
\(L(\frac12)\) \(\approx\) \(0.8794034044 + 1.726450708i\)
\(L(1)\) \(\approx\) \(1.053041114 + 0.8547082383i\)
\(L(1)\) \(\approx\) \(1.053041114 + 0.8547082383i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.649 + 0.760i)T \)
3 \( 1 + (-0.999 - 0.0119i)T \)
5 \( 1 + (0.385 + 0.922i)T \)
7 \( 1 + (0.996 - 0.0838i)T \)
11 \( 1 + (0.295 - 0.955i)T \)
13 \( 1 + (-0.107 + 0.994i)T \)
17 \( 1 + (0.940 - 0.340i)T \)
19 \( 1 + (-0.249 + 0.968i)T \)
23 \( 1 + (0.913 - 0.407i)T \)
29 \( 1 + (0.838 - 0.544i)T \)
31 \( 1 + (0.790 + 0.612i)T \)
37 \( 1 + (0.999 + 0.0359i)T \)
41 \( 1 + (-0.968 - 0.249i)T \)
43 \( 1 + (0.981 + 0.190i)T \)
47 \( 1 + (0.0239 - 0.999i)T \)
53 \( 1 + (-0.513 + 0.857i)T \)
59 \( 1 + (0.702 + 0.711i)T \)
61 \( 1 + (0.493 - 0.869i)T \)
67 \( 1 + (-0.744 + 0.667i)T \)
71 \( 1 + (0.640 - 0.767i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (-0.493 - 0.869i)T \)
83 \( 1 + (-0.931 - 0.363i)T \)
89 \( 1 + (-0.996 - 0.0838i)T \)
97 \( 1 + (-0.554 + 0.832i)T \)
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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

−21.17931166844481062478142028751, −20.78142748193481376123808411655, −19.936515151923938127665948669993, −19.03333196112595179368872542959, −17.78382955327744473731526509719, −17.674200392804423318217343613018, −16.76125052759961306095278275801, −15.54790931141073412573242338261, −15.02740411966393952027021400162, −14.00743181970206497915960109301, −12.91029996453330189407119480827, −12.625871889826572574493906595405, −11.76027342568428556743989566332, −11.10968187367655312611387303848, −10.16208015382493507464462184367, −9.59850752485876395052390066608, −8.451659791853970378878721856010, −7.29476697503255354972588739996, −6.145262108957101732978844227104, −5.29969549662198089975059844470, −4.84032427587760299676189183142, −4.15669276019779052752021893022, −2.64349572790468920720499928389, −1.472165732362792037273985306115, −0.912207884123349736741479647808, 1.28391789261904340882203673374, 2.631194414333381264158454599669, 3.79530358084146064174235989912, 4.64807511627346211515209225573, 5.54773966145125457402090053096, 6.21835479584259412383794615372, 6.92381937505734551996076857653, 7.732068331825442198031197293420, 8.70947811279686599298300642666, 9.95270697415832772369099235406, 10.87298884833426071781119060071, 11.644802120031069713542687569589, 12.09178115952013052892082266883, 13.3446636213835544083728898667, 14.15587253244760539905941409054, 14.53656178151367280704882224256, 15.52887589771250309215420542892, 16.5055632902993764045348043571, 16.976686576831196954306527984065, 17.68617217686024875642418984275, 18.57854580774854279438407340600, 18.98753564608060853251036892673, 20.84922911228093595042226184095, 21.38357765697256071038439363324, 21.78213028940676603057526931416

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