↑

Properties

Label 1-1033-1033.1002-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.937 - 0.347i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (−0.992 − 0.121i)2-s + (0.820 + 0.571i)3-s + (0.970 + 0.241i)4-s + (0.264 − 0.964i)5-s + (−0.744 − 0.667i)6-s + (−0.391 + 0.920i)7-s + (−0.934 − 0.357i)8-s + (0.345 + 0.938i)9-s + (−0.379 + 0.925i)10-s + (0.531 − 0.847i)11-s + (0.658 + 0.752i)12-s + (−0.938 + 0.345i)13-s + (0.5 − 0.866i)14-s + (0.768 − 0.639i)15-s + (0.883 + 0.468i)16-s + (−0.995 − 0.0972i)17-s + ⋯
L(s)  = 1  + (−0.992 − 0.121i)2-s + (0.820 + 0.571i)3-s + (0.970 + 0.241i)4-s + (0.264 − 0.964i)5-s + (−0.744 − 0.667i)6-s + (−0.391 + 0.920i)7-s + (−0.934 − 0.357i)8-s + (0.345 + 0.938i)9-s + (−0.379 + 0.925i)10-s + (0.531 − 0.847i)11-s + (0.658 + 0.752i)12-s + (−0.938 + 0.345i)13-s + (0.5 − 0.866i)14-s + (0.768 − 0.639i)15-s + (0.883 + 0.468i)16-s + (−0.995 − 0.0972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1002, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01469312915 - 0.08199882416i\)
\(L(\frac12)\) \(\approx\) \(0.01469312915 - 0.08199882416i\)
\(L(1)\) \(\approx\) \(0.6625554730 + 0.02811417514i\)
\(L(1)\) \(\approx\) \(0.6625554730 + 0.02811417514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.992 - 0.121i)T \)
3 \( 1 + (0.820 + 0.571i)T \)
5 \( 1 + (0.264 - 0.964i)T \)
7 \( 1 + (-0.391 + 0.920i)T \)
11 \( 1 + (0.531 - 0.847i)T \)
13 \( 1 + (-0.938 + 0.345i)T \)
17 \( 1 + (-0.995 - 0.0972i)T \)
19 \( 1 + (-0.999 - 0.0243i)T \)
23 \( 1 + (-0.978 - 0.205i)T \)
29 \( 1 + (-0.783 - 0.620i)T \)
31 \( 1 + (-0.711 + 0.702i)T \)
37 \( 1 + (-0.0365 - 0.999i)T \)
41 \( 1 + (0.985 - 0.169i)T \)
43 \( 1 + (-0.561 + 0.827i)T \)
47 \( 1 + (-0.630 - 0.776i)T \)
53 \( 1 + (-0.967 + 0.252i)T \)
59 \( 1 + (0.783 - 0.620i)T \)
61 \( 1 + (-0.989 - 0.145i)T \)
67 \( 1 + (-0.973 + 0.229i)T \)
71 \( 1 + (-0.264 - 0.964i)T \)
73 \( 1 + (0.999 + 0.0365i)T \)
79 \( 1 + (0.468 + 0.883i)T \)
83 \( 1 + (-0.776 - 0.630i)T \)
89 \( 1 + (0.983 - 0.181i)T \)
97 \( 1 + (0.334 + 0.942i)T \)
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

−21.90480663524997087074795101935, −20.71091668717975708429320983909, −20.012100801346950007316869914967, −19.590495331738664693545787781380, −18.88252192035542573358443642726, −17.94606339424524698308966662258, −17.51444319587284501875825798177, −16.720057060384186641821306766575, −15.4503062879848221776937667319, −14.8550497820438828490171105572, −14.30137620948865903796792476744, −13.236984322731404098201768948636, −12.42969784671081838382901663848, −11.36182234263738691920703378602, −10.43870753800650768542555021003, −9.78348463480868168689667169260, −9.17313355359721093619827265671, −7.97378289523112997570469202006, −7.31641594413634852485832918313, −6.78180720494948761941478378610, −6.0964120395177155444695404025, −4.25931018384310430968300183226, −3.247037275871347537693137799922, −2.26174941023330808831911230187, −1.66102515281330199772342821271, 0.037629246498248030271449754641, 1.87527889960711858212870217597, 2.30965697718879951853237008877, 3.51265757105418679643021571983, 4.5259037615195947892297679471, 5.69853128706200678754901793608, 6.56903992979067257607541870694, 7.850074352502442925866406200475, 8.55814837588792542916742449574, 9.22063881728794288240937833469, 9.50171009912530112152087928198, 10.6055866308751529125679679704, 11.54338818509394475725948074906, 12.45015977998773743604826560783, 13.16117765967115114836061756254, 14.33836993941186228569498734042, 15.10945371869396124011836653624, 16.0247883123469908059774807592, 16.40664017952826013348115777452, 17.224778627686895971669587685459, 18.17369466031998120471574711657, 19.191628743018594663724451401298, 19.62134535097645454062804941455, 20.189221198146123890704818132727, 21.3591476404701964149919717875

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