↑

Properties

Label 1-1027-1027.1004-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.437 + 0.899i$
Analytic cond. $110.366$
Root an. cond. $110.366$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060456161 + 0.6631181947i\)
\(L(\frac12)\) \(\approx\) \(1.060456161 + 0.6631181947i\)
\(L(1)\) \(\approx\) \(0.8920453809 - 0.1072661695i\)
\(L(1)\) \(\approx\) \(0.8920453809 - 0.1072661695i\)

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 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.32535184231768858764605359324, −19.95878348139287147525257413976, −19.52955259287820584900883781145, −18.621702643506576832132298977360, −18.336814517682870413794807726503, −17.53482261028236823173844276880, −16.647404024893482492541093541633, −15.522557020349948966715159048432, −14.81226717981006572911240829378, −14.51668383668604702249397746890, −13.631194287671592857039185407191, −12.57668850312857518174998305981, −11.55284401378119525081892922753, −11.01661002197058121045758144247, −9.54812722218245928516812170017, −8.960653592316144088487794962932, −8.19273402388550486309152932806, −7.28675035160038134196520362310, −6.82815263201367521287282596975, −5.96266261978433030989074823082, −4.921248744537945584663827675006, −3.61295539611567967421256636968, −2.530052387451703847354349536346, −1.51492367573089524964881113346, −0.325205350815053091934924312150, 1.05237644773139307062939172851, 1.81079773802660641093118281695, 3.2914285729356074482621618485, 4.09180901335549340059064787046, 4.366366073363624052208255860325, 5.598166089006418207921980106249, 7.34765677197420624820984753676, 8.04907078476285757349829978946, 8.76256124345344348534008015330, 9.440557439241762629925544062, 10.302494236057348146924606913522, 11.05775187369141849867615786072, 11.745308218223743734717548236785, 12.742095639761070789978935482607, 13.48934197337246988683998292421, 14.4656704928351544675234674921, 15.139582360697801015028950839473, 16.49278818035415191738229435266, 16.8606854897557357167147028485, 17.283524051758714649157052727199, 18.80972358312684189428480108656, 19.38991059526458013617928409271, 20.15105936526077233641410243423, 20.63033385396132498056683288878, 21.171561679434775585431881619002

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