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Properties

Label 1-1033-1033.80-r1-0-0
Degree $1$
Conductor $1033$
Sign $0.997 + 0.0642i$
Analytic cond. $111.011$
Root an. cond. $111.011$
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  + (0.820 − 0.571i)2-s + (0.0972 − 0.995i)3-s + (0.345 − 0.938i)4-s + (−0.850 + 0.526i)5-s + (−0.489 − 0.872i)6-s + (0.424 + 0.905i)7-s + (−0.252 − 0.967i)8-s + (−0.981 − 0.193i)9-s + (−0.396 + 0.917i)10-s + (−0.429 − 0.902i)11-s + (−0.900 − 0.435i)12-s + (0.556 + 0.830i)13-s + (0.866 + 0.5i)14-s + (0.440 + 0.897i)15-s + (−0.760 − 0.648i)16-s + (−0.468 − 0.883i)17-s + ⋯
L(s)  = 1  + (0.820 − 0.571i)2-s + (0.0972 − 0.995i)3-s + (0.345 − 0.938i)4-s + (−0.850 + 0.526i)5-s + (−0.489 − 0.872i)6-s + (0.424 + 0.905i)7-s + (−0.252 − 0.967i)8-s + (−0.981 − 0.193i)9-s + (−0.396 + 0.917i)10-s + (−0.429 − 0.902i)11-s + (−0.900 − 0.435i)12-s + (0.556 + 0.830i)13-s + (0.866 + 0.5i)14-s + (0.440 + 0.897i)15-s + (−0.760 − 0.648i)16-s + (−0.468 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.997 + 0.0642i$
Analytic conductor: \(111.011\)
Root analytic conductor: \(111.011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (1:\ ),\ 0.997 + 0.0642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.650010291 + 0.05309278548i\)
\(L(\frac12)\) \(\approx\) \(1.650010291 + 0.05309278548i\)
\(L(1)\) \(\approx\) \(1.141155385 - 0.6225404125i\)
\(L(1)\) \(\approx\) \(1.141155385 - 0.6225404125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.820 - 0.571i)T \)
3 \( 1 + (0.0972 - 0.995i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (0.424 + 0.905i)T \)
11 \( 1 + (-0.429 - 0.902i)T \)
13 \( 1 + (0.556 + 0.830i)T \)
17 \( 1 + (-0.468 - 0.883i)T \)
19 \( 1 + (-0.992 + 0.121i)T \)
23 \( 1 + (0.969 - 0.247i)T \)
29 \( 1 + (-0.546 + 0.837i)T \)
31 \( 1 + (-0.685 + 0.728i)T \)
37 \( 1 + (0.181 - 0.983i)T \)
41 \( 1 + (-0.752 + 0.658i)T \)
43 \( 1 + (0.987 + 0.157i)T \)
47 \( 1 + (-0.869 + 0.494i)T \)
53 \( 1 + (0.880 + 0.473i)T \)
59 \( 1 + (0.837 - 0.546i)T \)
61 \( 1 + (0.667 + 0.744i)T \)
67 \( 1 + (-0.362 + 0.931i)T \)
71 \( 1 + (-0.526 + 0.850i)T \)
73 \( 1 + (0.823 + 0.566i)T \)
79 \( 1 + (0.996 + 0.0790i)T \)
83 \( 1 + (-0.264 - 0.964i)T \)
89 \( 1 + (0.991 + 0.127i)T \)
97 \( 1 + (0.795 - 0.606i)T \)
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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

−21.16374575530657426353679695736, −20.766900756234075058442473169353, −20.19467439991985668782636858198, −19.422638998889566237510535253951, −17.83774995018915527408074534785, −17.00432400678623425132853731875, −16.70352580820894006117036154525, −15.48044518169582829369319109400, −15.25458166904118773569922607302, −14.64494907771353703256352423242, −13.28894137619176035795928674137, −13.00404283514382292031838773549, −11.80104715743898936137367933277, −10.99646721559872043081572578681, −10.41398482052418798124962460710, −9.05284004482087190005204272565, −8.18085787433280871511377919532, −7.700013845304952434133087193026, −6.59737372430200598381514044856, −5.38668603249321738241940412743, −4.72463651407100954176200822278, −4.0038778621500269044441294827, −3.49279060591302266721413238848, −2.102317255838331292919525211405, −0.28120435749375621301953427450, 0.93176635376307706188894039198, 2.11399306293275400333719605641, 2.80839775162578530951034635201, 3.65973448883629809856566470044, 4.84696399231286325209034937773, 5.7740003788764686554431780247, 6.615682322863438078827298610029, 7.311223209965894286570941064472, 8.542464011357226072097788952020, 9.03597696722567151652345319382, 10.78118501299505579187277117443, 11.251970584912647625508097482186, 11.77601101871043075239606999408, 12.70043636375950364405465032629, 13.289911366423999052723739775586, 14.44269546411175797910382476767, 14.60886747134218493540732975262, 15.752045683223584813860467087957, 16.41436547118561082397333812528, 17.99298513066780179143200075466, 18.590678065378550049520305509280, 19.027101140143278360475304624872, 19.68334901790593970837240465712, 20.67642014286530437175451868761, 21.4008017934638870458915932018

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