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Properties

Label 1-1049-1049.1024-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.218 + 0.975i$
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  + (−0.976 + 0.214i)2-s + (−0.385 − 0.922i)3-s + (0.908 − 0.418i)4-s + (−0.472 + 0.881i)5-s + (0.574 + 0.818i)6-s + (0.363 + 0.931i)7-s + (−0.797 + 0.603i)8-s + (−0.702 + 0.711i)9-s + (0.272 − 0.962i)10-s + (0.450 + 0.892i)11-s + (−0.736 − 0.676i)12-s + (0.407 + 0.913i)13-s + (−0.554 − 0.832i)14-s + (0.995 + 0.0957i)15-s + (0.649 − 0.760i)16-s + (0.887 + 0.461i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.214i)2-s + (−0.385 − 0.922i)3-s + (0.908 − 0.418i)4-s + (−0.472 + 0.881i)5-s + (0.574 + 0.818i)6-s + (0.363 + 0.931i)7-s + (−0.797 + 0.603i)8-s + (−0.702 + 0.711i)9-s + (0.272 − 0.962i)10-s + (0.450 + 0.892i)11-s + (−0.736 − 0.676i)12-s + (0.407 + 0.913i)13-s + (−0.554 − 0.832i)14-s + (0.995 + 0.0957i)15-s + (0.649 − 0.760i)16-s + (0.887 + 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.218 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4715749786 + 0.5890883391i\)
\(L(\frac12)\) \(\approx\) \(0.4715749786 + 0.5890883391i\)
\(L(1)\) \(\approx\) \(0.6106171274 + 0.1854925160i\)
\(L(1)\) \(\approx\) \(0.6106171274 + 0.1854925160i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.976 + 0.214i)T \)
3 \( 1 + (-0.385 - 0.922i)T \)
5 \( 1 + (-0.472 + 0.881i)T \)
7 \( 1 + (0.363 + 0.931i)T \)
11 \( 1 + (0.450 + 0.892i)T \)
13 \( 1 + (0.407 + 0.913i)T \)
17 \( 1 + (0.887 + 0.461i)T \)
19 \( 1 + (-0.897 + 0.440i)T \)
23 \( 1 + (0.958 - 0.283i)T \)
29 \( 1 + (0.989 + 0.143i)T \)
31 \( 1 + (0.225 - 0.974i)T \)
37 \( 1 + (0.927 + 0.374i)T \)
41 \( 1 + (-0.897 + 0.440i)T \)
43 \( 1 + (0.998 - 0.0479i)T \)
47 \( 1 + (-0.702 + 0.711i)T \)
53 \( 1 + (0.864 - 0.503i)T \)
59 \( 1 + (-0.554 + 0.832i)T \)
61 \( 1 + (-0.965 - 0.260i)T \)
67 \( 1 + (0.838 + 0.544i)T \)
71 \( 1 + (0.574 - 0.818i)T \)
73 \( 1 + (-0.825 + 0.564i)T \)
79 \( 1 + (-0.965 + 0.260i)T \)
83 \( 1 + (-0.295 + 0.955i)T \)
89 \( 1 + (0.363 - 0.931i)T \)
97 \( 1 + (-0.513 - 0.857i)T \)
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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

−21.21086865215073164147682160928, −20.4352219545357775324515842671, −19.87478494841185469598275592452, −19.17006309475870949980499897799, −17.97609789593065465967660235938, −17.130459825076643245791541679898, −16.82996084023206759475945712591, −16.04174354201800313056509107256, −15.44288433306996303180063583646, −14.43779400553154122823761429196, −13.2654927202972822864719340744, −12.25776486338001786179979707349, −11.47539694809149218183497734565, −10.81650237396116294254942440231, −10.212293234106876920259220346304, −9.178967829236548535254767957685, −8.5492584110195774438065593041, −7.8513860418604534255663541949, −6.74125526492768811538534966548, −5.6756360200489914514875636401, −4.71666746831706157912434186269, −3.71883529784427729867876067892, −3.045004758898585461958947156909, −1.14973268210591125373824090834, −0.58189408415144742414292366186, 1.27973271737583385016588566511, 2.10893554134327177497402161255, 2.92032284741034936285003391805, 4.50439844431936844788035586283, 5.87816993101719698334661089627, 6.44056794668779658058425237423, 7.141421459209384787044431050207, 8.01117299921533464277409172247, 8.610955984220042000934950880485, 9.70265114927607781676502451735, 10.675901931451511600316712235552, 11.466148175785762573176364363921, 11.95310975708816123714429936315, 12.72723834519269344872987650493, 14.243337630734185244961514140069, 14.75790223077380327674097610048, 15.45535171574360680298310852962, 16.59061793022085308142267123466, 17.187941003290030143640381818707, 18.091627265753749673545982942049, 18.61549135233704363347312367061, 19.11151094244086670424656500755, 19.72151012098519934256074697730, 20.906709440364363552058994981564, 21.683171702192221812696643578860

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