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Properties

Label 1-1045-1045.653-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.953 - 0.299i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.207 + 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (−0.978 + 0.207i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (0.743 + 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.587 − 0.809i)18-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s + (0.978 + 0.207i)24-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (−0.978 + 0.207i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (0.743 + 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.587 − 0.809i)18-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s + (0.978 + 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.953 - 0.299i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.953 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3362957078 + 2.191424621i\)
\(L(\frac12)\) \(\approx\) \(-0.3362957078 + 2.191424621i\)
\(L(1)\) \(\approx\) \(0.6834392006 + 0.9421517114i\)
\(L(1)\) \(\approx\) \(0.6834392006 + 0.9421517114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.994 + 0.104i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−20.83239898676490889978745267429, −20.04155019940364154306980749231, −19.50635626592280649483309069765, −18.59991447650495697133980646909, −18.06833202641661390711496741114, −17.26333289509228626662567336441, −16.66775303781965475310372541209, −15.05326170112221552235072763989, −14.406150713956366163725201212395, −13.46048102881862537806952248874, −13.08380659921671621463001571072, −12.22994397454045189078350222801, −11.24892936598894962952942476930, −10.76856626398701748057954244132, −9.67294867195733924950720817141, −8.80731820064809771717733705896, −7.862202134537288954331641019968, −7.5652910525307428845899516836, −6.165015172599693070734956680597, −5.14363668834272022238006217452, −3.86389161559419980685698327034, −3.290423591275148870891251675720, −2.10716585330856343223124021618, −1.24160957610705331014336130829, −0.55676082725875511718447668334, 1.133507091289897461514957939856, 2.52447453115465718178927431375, 3.69535218024325438731474163894, 4.62760641673779236138078963723, 5.319869056448655283027710592611, 6.10353050776389125480046308067, 7.28385452954038787476187235963, 8.18172600624457292990090481923, 8.91172768647187121455535867037, 9.363967629246084495232535324, 10.405847675839315905623483915881, 11.23932671566913960009716857572, 12.26414697639633555634740697179, 13.5729500921745281158892384336, 14.08250189750404503587775010085, 15.01646802614596202181971360547, 15.330879206718973005064115329442, 16.36970046324213331320103253585, 16.727902940063377519023903496056, 17.815322093030593626704875899518, 18.648257359068265058724580048450, 19.1482832328881724483583929045, 20.37733754766866529816819262944, 21.13179297870539802648989870139, 21.738351045632417596547819813974

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