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Properties

Label 1-1063-1063.37-r0-0-0
Degree $1$
Conductor $1063$
Sign $-0.502 - 0.864i$
Analytic cond. $4.93655$
Root an. cond. $4.93655$
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  + (0.603 + 0.797i)2-s + (−0.870 + 0.492i)3-s + (−0.271 + 0.962i)4-s + (−0.887 − 0.461i)5-s + (−0.917 − 0.396i)6-s + (−0.5 + 0.866i)7-s + (−0.931 + 0.364i)8-s + (0.515 − 0.857i)9-s + (−0.167 − 0.985i)10-s + (0.421 + 0.906i)11-s + (−0.237 − 0.971i)12-s + (0.949 + 0.314i)13-s + (−0.992 + 0.123i)14-s + (0.999 − 0.0354i)15-s + (−0.852 − 0.522i)16-s + (−0.132 + 0.991i)17-s + ⋯
L(s)  = 1  + (0.603 + 0.797i)2-s + (−0.870 + 0.492i)3-s + (−0.271 + 0.962i)4-s + (−0.887 − 0.461i)5-s + (−0.917 − 0.396i)6-s + (−0.5 + 0.866i)7-s + (−0.931 + 0.364i)8-s + (0.515 − 0.857i)9-s + (−0.167 − 0.985i)10-s + (0.421 + 0.906i)11-s + (−0.237 − 0.971i)12-s + (0.949 + 0.314i)13-s + (−0.992 + 0.123i)14-s + (0.999 − 0.0354i)15-s + (−0.852 − 0.522i)16-s + (−0.132 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ -0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4205751686 + 0.7306141234i\)
\(L(\frac12)\) \(\approx\) \(-0.4205751686 + 0.7306141234i\)
\(L(1)\) \(\approx\) \(0.4843302532 + 0.6826123690i\)
\(L(1)\) \(\approx\) \(0.4843302532 + 0.6826123690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.603 + 0.797i)T \)
3 \( 1 + (-0.870 + 0.492i)T \)
5 \( 1 + (-0.887 - 0.461i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.421 + 0.906i)T \)
13 \( 1 + (0.949 + 0.314i)T \)
17 \( 1 + (-0.132 + 0.991i)T \)
19 \( 1 + (0.150 + 0.988i)T \)
23 \( 1 + (0.220 + 0.975i)T \)
29 \( 1 + (0.823 + 0.567i)T \)
31 \( 1 + (-0.870 + 0.492i)T \)
37 \( 1 + (-0.697 - 0.716i)T \)
41 \( 1 + (0.185 + 0.982i)T \)
43 \( 1 + (-0.931 - 0.364i)T \)
47 \( 1 + (0.115 - 0.993i)T \)
53 \( 1 + (-0.954 - 0.297i)T \)
59 \( 1 + (0.994 + 0.106i)T \)
61 \( 1 + (0.895 - 0.445i)T \)
67 \( 1 + (0.00887 + 0.999i)T \)
71 \( 1 + (0.574 + 0.818i)T \)
73 \( 1 + (-0.0620 - 0.998i)T \)
79 \( 1 + (0.879 + 0.476i)T \)
83 \( 1 + (-0.973 - 0.228i)T \)
89 \( 1 + (0.658 - 0.752i)T \)
97 \( 1 + (0.758 + 0.651i)T \)
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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

−20.94808076734666612109110750226, −20.12960971467442198147677808157, −19.415109259958092718431122401347, −18.79006118181028317771117412489, −18.18582071451368065615751219340, −17.14749108376871149732754804454, −16.04897205439826717667402409075, −15.7336143687821008187776214029, −14.380438550840909950833994515593, −13.63275308654306093187472632709, −13.08278338684359403756575179369, −12.11790410670925084918581451258, −11.35976502720870481591405358466, −10.941392489188041780963789146101, −10.26638096435444961149975207339, −9.03886123951777470184423089591, −7.89311471076521985848061087533, −6.69862307112724044463765403427, −6.45693969099386848206413621363, −5.20302910208615252234031313506, −4.30895706633225103115023435551, −3.49093973767248255067123449690, −2.63270789136250673904450908723, −1.02969681632668018865806921485, −0.406639889052312177310607916424, 1.59219356635681612273139478805, 3.5353798938621666761758728455, 3.81211528007060663022022327514, 4.90997244089168018837180500471, 5.57758857642422697895392418122, 6.45000279224427847944802289113, 7.14500786843116237700841645393, 8.36183038723529691125020619156, 8.96926907098072765013273172239, 9.94059821702544150146556259943, 11.222917934051099631399920402180, 11.9295704159004098032983694799, 12.50447422190080545917533593348, 13.0774228138053647342043357406, 14.549446378913001219582953818844, 15.14065119343789797297521473782, 15.91086183799243602511125303522, 16.2266476974133002126976847426, 17.10556781228900915503381255719, 17.90187961083870653218394868659, 18.69055318852160308985797036841, 19.78123070387052300136538702873, 20.76050719569610395845632608791, 21.50691265396127999776751650695, 22.12672100380590269486832360310

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