↑

Properties

Label 1-1033-1033.1026-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.653 + 0.757i$
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  + (−0.181 + 0.983i)2-s + (0.791 − 0.611i)3-s + (−0.934 − 0.357i)4-s + (−0.920 − 0.391i)5-s + (0.457 + 0.889i)6-s + (0.181 + 0.983i)7-s + (0.520 − 0.853i)8-s + (0.252 − 0.967i)9-s + (0.551 − 0.833i)10-s + (−0.667 − 0.744i)11-s + (−0.957 + 0.288i)12-s + (−0.967 − 0.252i)13-s − 14-s + (−0.967 + 0.252i)15-s + (0.744 + 0.667i)16-s + (−0.989 − 0.145i)17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (0.791 − 0.611i)3-s + (−0.934 − 0.357i)4-s + (−0.920 − 0.391i)5-s + (0.457 + 0.889i)6-s + (0.181 + 0.983i)7-s + (0.520 − 0.853i)8-s + (0.252 − 0.967i)9-s + (0.551 − 0.833i)10-s + (−0.667 − 0.744i)11-s + (−0.957 + 0.288i)12-s + (−0.967 − 0.252i)13-s − 14-s + (−0.967 + 0.252i)15-s + (0.744 + 0.667i)16-s + (−0.989 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.653 + 0.757i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.653 + 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2821480427 + 0.6162416027i\)
\(L(\frac12)\) \(\approx\) \(0.2821480427 + 0.6162416027i\)
\(L(1)\) \(\approx\) \(0.7599266551 + 0.2573520150i\)
\(L(1)\) \(\approx\) \(0.7599266551 + 0.2573520150i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.181 + 0.983i)T \)
3 \( 1 + (0.791 - 0.611i)T \)
5 \( 1 + (-0.920 - 0.391i)T \)
7 \( 1 + (0.181 + 0.983i)T \)
11 \( 1 + (-0.667 - 0.744i)T \)
13 \( 1 + (-0.967 - 0.252i)T \)
17 \( 1 + (-0.989 - 0.145i)T \)
19 \( 1 + (-0.0365 + 0.999i)T \)
23 \( 1 + (0.889 - 0.457i)T \)
29 \( 1 + (-0.217 + 0.976i)T \)
31 \( 1 + (-0.391 + 0.920i)T \)
37 \( 1 + (0.744 + 0.667i)T \)
41 \( 1 + (-0.252 - 0.967i)T \)
43 \( 1 + (-0.109 + 0.994i)T \)
47 \( 1 + (0.853 + 0.520i)T \)
53 \( 1 + (-0.920 - 0.391i)T \)
59 \( 1 + (0.217 + 0.976i)T \)
61 \( 1 + (0.976 + 0.217i)T \)
67 \( 1 + (-0.424 + 0.905i)T \)
71 \( 1 + (0.920 - 0.391i)T \)
73 \( 1 + (0.667 + 0.744i)T \)
79 \( 1 + (-0.667 - 0.744i)T \)
83 \( 1 + (-0.520 - 0.853i)T \)
89 \( 1 + (-0.489 + 0.872i)T \)
97 \( 1 + (0.489 + 0.872i)T \)
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

−21.10412506609977425902859610402, −20.24839480864921502400510440967, −19.92051847959290644895228415482, −19.28963482804552632013080938306, −18.46433364836477237863651694508, −17.40385732297062576839507736943, −16.774468145675384898436848161223, −15.502166216429088347198435415965, −15.03610754256828068906323469938, −14.10876397877784856236009620271, −13.29851357519099003903273768687, −12.64657152214136901006907530728, −11.24599820781757135367782265776, −11.06973026288835064866535443962, −10.02077692953853371261364387052, −9.46740047596163293663681095394, −8.43115961783682589587841440963, −7.57814677765792432168012940774, −7.12764620720272195845948708494, −4.94192546609979316569387575429, −4.44144281398951029482316667622, −3.75518228215548922841730324818, −2.72359886659333123482353910728, −2.05934423522591985121815691416, −0.30274246041354538103737013372, 1.137469823684490543708921302610, 2.55221280018279168432284086172, 3.50672080571491613333004328009, 4.69219764219056927600320415283, 5.44557023110486387587775195380, 6.54747652763305406391026763956, 7.41176068196922871822904916703, 8.095761053103548082524674927461, 8.71627990788454331412225004846, 9.24007557337630290077677249442, 10.52481280162711648351127269444, 11.73602376782890748581356638964, 12.72050388417801806665152857675, 13.01998634321656200894229379010, 14.33595852907570839364043877495, 14.77187469093682835325420475282, 15.58472008869846649384313338719, 16.11954110453548528790966460409, 17.140699311872972592056988465282, 18.146024985410125994171207664283, 18.69650016221112295118018256091, 19.281915534105595616578131769, 20.05523953885506443038335453679, 21.018142449895006406913262848982, 22.00326478375355206494059344606

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