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Properties

Label 1-1063-1063.1022-r0-0-0
Degree $1$
Conductor $1063$
Sign $0.814 + 0.580i$
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  + (0.937 − 0.347i)2-s + (0.603 + 0.797i)3-s + (0.758 − 0.651i)4-s + (0.910 − 0.413i)5-s + (0.842 + 0.537i)6-s + (−0.5 + 0.866i)7-s + (0.484 − 0.874i)8-s + (−0.271 + 0.962i)9-s + (0.710 − 0.703i)10-s + (−0.981 + 0.194i)11-s + (0.977 + 0.211i)12-s + (−0.237 + 0.971i)13-s + (−0.167 + 0.985i)14-s + (0.879 + 0.476i)15-s + (0.150 − 0.988i)16-s + (0.288 + 0.957i)17-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)2-s + (0.603 + 0.797i)3-s + (0.758 − 0.651i)4-s + (0.910 − 0.413i)5-s + (0.842 + 0.537i)6-s + (−0.5 + 0.866i)7-s + (0.484 − 0.874i)8-s + (−0.271 + 0.962i)9-s + (0.710 − 0.703i)10-s + (−0.981 + 0.194i)11-s + (0.977 + 0.211i)12-s + (−0.237 + 0.971i)13-s + (−0.167 + 0.985i)14-s + (0.879 + 0.476i)15-s + (0.150 − 0.988i)16-s + (0.288 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.814 + 0.580i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ 0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.488862822 + 1.115571306i\)
\(L(\frac12)\) \(\approx\) \(3.488862822 + 1.115571306i\)
\(L(1)\) \(\approx\) \(2.335952258 + 0.2949452313i\)
\(L(1)\) \(\approx\) \(2.335952258 + 0.2949452313i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.937 - 0.347i)T \)
3 \( 1 + (0.603 + 0.797i)T \)
5 \( 1 + (0.910 - 0.413i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.981 + 0.194i)T \)
13 \( 1 + (-0.237 + 0.971i)T \)
17 \( 1 + (0.288 + 0.957i)T \)
19 \( 1 + (0.515 - 0.857i)T \)
23 \( 1 + (0.999 - 0.0354i)T \)
29 \( 1 + (-0.560 - 0.828i)T \)
31 \( 1 + (0.603 + 0.797i)T \)
37 \( 1 + (0.185 + 0.982i)T \)
41 \( 1 + (0.861 - 0.507i)T \)
43 \( 1 + (0.484 + 0.874i)T \)
47 \( 1 + (0.0443 + 0.999i)T \)
53 \( 1 + (-0.468 + 0.883i)T \)
59 \( 1 + (0.0797 - 0.996i)T \)
61 \( 1 + (0.984 + 0.176i)T \)
67 \( 1 + (-0.992 - 0.123i)T \)
71 \( 1 + (0.658 - 0.752i)T \)
73 \( 1 + (-0.645 - 0.764i)T \)
79 \( 1 + (0.781 - 0.624i)T \)
83 \( 1 + (-0.996 + 0.0886i)T \)
89 \( 1 + (0.802 - 0.596i)T \)
97 \( 1 + (-0.870 + 0.492i)T \)
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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

−21.280460916764453876519972626753, −20.664324487730467713756334594551, −20.20114848847703011353797523620, −19.107454161108230684063269584, −18.251365461346120309204813444097, −17.54223140447348207740869540637, −16.68770004974601933327340214026, −15.82702489224154415351654480208, −14.79116078555628187037394971710, −14.25959964959209222700902386979, −13.38402431411384937235541285305, −13.14802751788377258242723119573, −12.352878205176809429941034543732, −11.14491964389429641028944144293, −10.30654185588683527325440665483, −9.41828150167826955246332772789, −8.12028150147606568571412825836, −7.36151352798706929560739590774, −6.91582832563593262952353048852, −5.79011254966613042727255788700, −5.28658049625418744198103067347, −3.75977161364199104985794119956, −2.95007483571557178519657223917, −2.43064766326165536415216862779, −1.04752321385473421433721330093, 1.59773577485441026388090953259, 2.56818466723891280743451413149, 2.998302173316498972546557349759, 4.351036441094351408097463084428, 5.00738301405166433034782488009, 5.72860020116211212286501757956, 6.60911848745903940870324014629, 7.89747600526991256467723047118, 9.11533903094863722257605387256, 9.54459693533165365530281113808, 10.38630256438002115374012010104, 11.19267080354339665929718591302, 12.30287752025671632867791989652, 13.057472457761275621933078850, 13.617189603917804338993149156425, 14.48992553578243181472787354693, 15.21144656521914289111989868223, 15.88228334791443756756919078608, 16.560474476394675642809987666689, 17.59288136864077236493443266554, 18.95761631555182027479810003546, 19.26775586394134367226043816069, 20.39248893714591947827140187009, 21.04738382179252904377272111549, 21.47762086733218754181389806279

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