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Properties

Label 1-1049-1049.1000-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.945 - 0.326i$
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  + (0.971 + 0.237i)2-s + (0.996 − 0.0838i)3-s + (0.887 + 0.461i)4-s + (−0.363 − 0.931i)5-s + (0.987 + 0.155i)6-s + (−0.832 − 0.554i)7-s + (0.752 + 0.658i)8-s + (0.985 − 0.167i)9-s + (−0.131 − 0.991i)10-s + (−0.864 + 0.503i)11-s + (0.922 + 0.385i)12-s + (0.685 + 0.727i)13-s + (−0.676 − 0.736i)14-s + (−0.440 − 0.897i)15-s + (0.574 + 0.818i)16-s + (0.760 − 0.649i)17-s + ⋯
L(s)  = 1  + (0.971 + 0.237i)2-s + (0.996 − 0.0838i)3-s + (0.887 + 0.461i)4-s + (−0.363 − 0.931i)5-s + (0.987 + 0.155i)6-s + (−0.832 − 0.554i)7-s + (0.752 + 0.658i)8-s + (0.985 − 0.167i)9-s + (−0.131 − 0.991i)10-s + (−0.864 + 0.503i)11-s + (0.922 + 0.385i)12-s + (0.685 + 0.727i)13-s + (−0.676 − 0.736i)14-s + (−0.440 − 0.897i)15-s + (0.574 + 0.818i)16-s + (0.760 − 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.945 - 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.684445121 - 0.6181777280i\)
\(L(\frac12)\) \(\approx\) \(3.684445121 - 0.6181777280i\)
\(L(1)\) \(\approx\) \(2.395429658 - 0.1315892110i\)
\(L(1)\) \(\approx\) \(2.395429658 - 0.1315892110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.971 + 0.237i)T \)
3 \( 1 + (0.996 - 0.0838i)T \)
5 \( 1 + (-0.363 - 0.931i)T \)
7 \( 1 + (-0.832 - 0.554i)T \)
11 \( 1 + (-0.864 + 0.503i)T \)
13 \( 1 + (0.685 + 0.727i)T \)
17 \( 1 + (0.760 - 0.649i)T \)
19 \( 1 + (0.981 - 0.190i)T \)
23 \( 1 + (0.979 - 0.202i)T \)
29 \( 1 + (-0.631 - 0.775i)T \)
31 \( 1 + (0.0957 - 0.995i)T \)
37 \( 1 + (-0.968 + 0.249i)T \)
41 \( 1 + (-0.190 - 0.981i)T \)
43 \( 1 + (0.225 - 0.974i)T \)
47 \( 1 + (0.167 + 0.985i)T \)
53 \( 1 + (-0.593 - 0.804i)T \)
59 \( 1 + (0.736 + 0.676i)T \)
61 \( 1 + (0.450 + 0.892i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (-0.987 + 0.155i)T \)
73 \( 1 + (-0.472 + 0.881i)T \)
79 \( 1 + (-0.450 + 0.892i)T \)
83 \( 1 + (-0.857 - 0.513i)T \)
89 \( 1 + (0.832 - 0.554i)T \)
97 \( 1 + (-0.825 - 0.564i)T \)
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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

−21.65383378424315664515590081659, −20.8638268043432661128212333566, −20.12131152363886840265315701945, −19.22646933276607999670719855864, −18.866092367446978514714534526504, −18.090889023632104124048389875069, −16.25734804555233862760647326845, −15.84905641977866785674697583662, −15.13710086506182714663645135373, −14.53346009322582763448810971390, −13.6678020202712410427937533575, −13.0008121639524885281926633714, −12.345256103741732606276176590, −11.17997254325187744840256029160, −10.4406643082954946052048013397, −9.81193832226092724419199842544, −8.56287033649020484413470096412, −7.65679848431182958520444908543, −6.91044485367664873170464906363, −5.91630561197716003264724341074, −5.11759867454084789557800274928, −3.590817884157144413193078856335, −3.25501220288538817145909361535, −2.74388505096636024230782779194, −1.42590311939036124907229063441, 1.122175594340177644930953218383, 2.33883393717237223572130774471, 3.32236926320139846700365411695, 3.98207370021817241413355791315, 4.82586162899792343533005969023, 5.76800980530274500600767843735, 7.15921345116603320073393272380, 7.39755569884955083473333982753, 8.44437232287560657443508156875, 9.363762236876279801962596585408, 10.18265911122838340924210755299, 11.42436790889784580471267287162, 12.31045690207550360230258381915, 13.06880008811075523278839108447, 13.50193053234674975591795524389, 14.193716384306557690249195983349, 15.34436169565030003536035815287, 15.834209547606581570339811040248, 16.4121945031802256678302289022, 17.30422936130167097946394658672, 18.769417940871249702333306197471, 19.28252885754729854202258734801, 20.44379075861078196706223073369, 20.60316986348208974843386516708, 21.13731042435497197598205265684

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