↑

Properties

Label 1-1035-1035.229-r1-0-0
Degree $1$
Conductor $1035$
Sign $-0.939 + 0.342i$
Analytic cond. $111.226$
Root an. cond. $111.226$
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)4-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)22-s + 26-s + 28-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)22-s + 26-s + 28-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(111.226\)
Root analytic conductor: \(111.226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1035,\ (1:\ ),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1557013901 - 0.8830264634i\)
\(L(\frac12)\) \(\approx\) \(-0.1557013901 - 0.8830264634i\)
\(L(1)\) \(\approx\) \(0.9515396564 - 0.5263246529i\)
\(L(1)\) \(\approx\) \(0.9515396564 - 0.5263246529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
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

−22.04200754331603171921996457827, −21.07645053412629590386232021914, −20.38213047259043451654293835061, −19.55211779297687695627005555061, −18.48985406972895602366029070096, −17.66325296517357872560626874457, −16.92019413805895762441606610960, −16.43815514155378911630042834358, −15.38641209810610621576783962662, −14.8208543298264756548105187896, −13.99186187935968677371499705243, −13.11290045278013485628252302360, −12.6377879684937488404083332244, −11.67542634522386686152694944097, −10.457129441496358953377645992157, −9.74234665316441581287231991545, −8.71454908922097212707270222355, −7.7841859099027844463431205059, −7.12699626156771288425572505911, −6.2878758223190028089651340618, −5.48428150678558942498562259938, −4.33774416291001266088137922308, −3.767476777426391075322456870595, −2.750824261623596458342075980241, −1.13657785620449868747213175158, 0.1655731646356914732926880337, 1.42440921819545802476430235668, 2.35551448825742439692694177896, 3.39536624899894720140466279862, 4.010097732768251831182623065, 5.26137913199392919629138572217, 5.98951818326313290017630836358, 6.68638972203560528522120362919, 8.28455622253822240844139570753, 9.04994320084813660184916947379, 9.648602832065484826364710590020, 10.74560224685126826528552798098, 11.45068575985806268586589872029, 12.158966126041828993496710622704, 12.91707554443736648267016964249, 13.70117870281263467182978943510, 14.550474083641451173559883692052, 15.16344621563600845879721949917, 16.27506118343226921379191497386, 16.90099104332555333436273420408, 18.32507265472795156225383735226, 18.75491608671890747272569781663, 19.31728305972233475076972075891, 20.190321036192470171149510706428, 21.170728775433724620028300938662

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