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Properties

Label 1-1027-1027.529-r0-0-0
Degree $1$
Conductor $1027$
Sign $-0.777 - 0.628i$
Analytic cond. $4.76936$
Root an. cond. $4.76936$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(4.76936\)
Root analytic conductor: \(4.76936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (0:\ ),\ -0.777 - 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1224247765 + 0.3460310091i\)
\(L(\frac12)\) \(\approx\) \(-0.1224247765 + 0.3460310091i\)
\(L(1)\) \(\approx\) \(0.3982294028 + 0.3611193758i\)
\(L(1)\) \(\approx\) \(0.3982294028 + 0.3611193758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.97921289327010677381371975549, −20.12813780712958822518182830890, −19.32045442653645406893886630896, −19.05310237445316192109143446832, −18.093666655487452808566110385692, −17.36545637721611073958088902861, −16.56358827581496083843388477051, −16.06819244425477443746668199337, −14.708597058758830762014455544196, −13.52965798615151260665866572504, −12.85859635320968611729222498690, −12.2093401680351459167572737695, −11.72329505737781948211751911127, −11.01773426527712078255759728512, −9.72777445086274085187861344986, −8.92779105838621728442400774073, −8.39104935237622433738360353821, −7.38127089989200515320563240530, −6.473316417455602677998072071679, −5.348565203547375697869526244328, −4.442946947709626187213821713492, −3.323198225136784615772899793134, −2.221534293633340079620881718, −1.3386638270328411564498311043, −0.24198689204063049279303105195, 1.13759011416925982918846057315, 3.04703352629600317836274934051, 4.06886708323597637974920468449, 4.59247688592934330605088643961, 5.9193836725232419451495889282, 6.77106244031290235836208282510, 7.01223653984193377795935117371, 8.4386417387556755338742253373, 9.14615912158574534400760350908, 10.10273856265062282940800979002, 10.7233595161925308264772903898, 11.24403317503259020666291056965, 12.51094951955189075272164502401, 13.75450537385609588510651569010, 14.47861590129481712700395940914, 15.19547027821940489243888871546, 15.77506117416946489484004691215, 16.59338648833427279704013220645, 17.37710686105963878056216013801, 17.688660067662221087905626222963, 19.07700178115768280729533055330, 19.505507240437085350413208833251, 20.25630210916230104577083809406, 21.589079768157577489988893780345, 22.336448599759245745359691794802

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