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Properties

Label 1-1063-1063.56-r0-0-0
Degree $1$
Conductor $1063$
Sign $0.0104 + 0.999i$
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  + (0.999 − 0.0118i)2-s + (0.900 + 0.434i)3-s + (0.999 − 0.0236i)4-s + (0.322 + 0.946i)5-s + (0.905 + 0.423i)6-s + (−0.939 + 0.342i)7-s + (0.999 − 0.0354i)8-s + (0.622 + 0.782i)9-s + (0.333 + 0.942i)10-s + (−0.738 + 0.673i)11-s + (0.910 + 0.413i)12-s + (0.842 − 0.537i)13-s + (−0.935 + 0.353i)14-s + (−0.120 + 0.992i)15-s + (0.998 − 0.0473i)16-s + (−0.833 + 0.552i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0118i)2-s + (0.900 + 0.434i)3-s + (0.999 − 0.0236i)4-s + (0.322 + 0.946i)5-s + (0.905 + 0.423i)6-s + (−0.939 + 0.342i)7-s + (0.999 − 0.0354i)8-s + (0.622 + 0.782i)9-s + (0.333 + 0.942i)10-s + (−0.738 + 0.673i)11-s + (0.910 + 0.413i)12-s + (0.842 − 0.537i)13-s + (−0.935 + 0.353i)14-s + (−0.120 + 0.992i)15-s + (0.998 − 0.0473i)16-s + (−0.833 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0104 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0104 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.0104 + 0.999i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ 0.0104 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.657423831 + 2.685445065i\)
\(L(\frac12)\) \(\approx\) \(2.657423831 + 2.685445065i\)
\(L(1)\) \(\approx\) \(2.226601766 + 1.008784338i\)
\(L(1)\) \(\approx\) \(2.226601766 + 1.008784338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.999 - 0.0118i)T \)
3 \( 1 + (0.900 + 0.434i)T \)
5 \( 1 + (0.322 + 0.946i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.738 + 0.673i)T \)
13 \( 1 + (0.842 - 0.537i)T \)
17 \( 1 + (-0.833 + 0.552i)T \)
19 \( 1 + (-0.926 - 0.375i)T \)
23 \( 1 + (0.809 + 0.586i)T \)
29 \( 1 + (-0.881 + 0.471i)T \)
31 \( 1 + (-0.0738 - 0.997i)T \)
37 \( 1 + (0.150 - 0.988i)T \)
41 \( 1 + (0.515 + 0.857i)T \)
43 \( 1 + (-0.468 - 0.883i)T \)
47 \( 1 + (0.966 + 0.257i)T \)
53 \( 1 + (0.718 - 0.695i)T \)
59 \( 1 + (0.631 - 0.775i)T \)
61 \( 1 + (-0.999 - 0.00591i)T \)
67 \( 1 + (0.584 + 0.811i)T \)
71 \( 1 + (-0.132 - 0.991i)T \)
73 \( 1 + (-0.179 - 0.983i)T \)
79 \( 1 + (0.126 + 0.991i)T \)
83 \( 1 + (0.867 + 0.497i)T \)
89 \( 1 + (0.684 + 0.728i)T \)
97 \( 1 + (-0.754 - 0.656i)T \)
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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

−21.15577124182992536935611278157, −20.62320346537182340118673205842, −20.00295671289761097086754224557, −19.16136773453533336789595584421, −18.516540212863215247247447425, −17.13832418001330993297254405041, −16.32108702077253505752366517781, −15.807713537732598175887900068541, −14.94765953689880979385332522001, −13.76762527841492665633632384829, −13.49312885977824535859677725328, −12.90346412640269854027155627395, −12.23396453880620315343219033655, −11.05185195903630799030939313142, −10.152244969029357977604965363041, −9.02496296357355047402462104184, −8.469630549223697134208760013157, −7.35642100648813467998670156717, −6.50659421640016417530584206618, −5.836469838366015728173320328684, −4.60230936726677055033245391559, −3.865213865989378418415395121410, −2.93447447923858706600070567746, −2.09239828120357718867548740033, −0.95753089213144562102846966288, 2.026057233224570177029013558379, 2.522129438428177256347822002291, 3.4274209908492309767789485102, 4.03406515136629952555665649887, 5.2628287931389761527581546519, 6.13381837111158795528607662862, 6.976342823137193274988759726241, 7.7288063077033855164474613568, 8.916771813826930848421459862139, 9.84798499279701412874381472280, 10.67661862394206353921164416700, 11.11214606334187006198399031634, 12.70166388200670950022182069199, 13.146865095732568027608237804677, 13.67340281163797591679990214502, 14.93738366050564500328673987236, 15.144343678137251533128944147200, 15.72954624198286602115345174040, 16.72242834822363434587177620209, 17.92468523507837483870690959337, 18.89790801100787112952068193296, 19.48500444629773372502431190913, 20.31184582493455406553881470042, 21.051074144302840196160530343311, 21.74883406126667598192300204161

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