Properties

Degree 1
Conductor $ 3^{2} \cdot 5 \cdot 23 $
Sign $-0.939 - 0.342i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
\( \varepsilon \)  =  $-0.939 - 0.342i$
motivic weight  =  \(0\)
character  :  $\chi_{1035} (574, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1035,\ (1:\ ),\ -0.939 - 0.342i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1557013901 + 0.8830264634i$
$L(\frac12,\chi)$  $\approx$  $-0.1557013901 + 0.8830264634i$
$L(\chi,1)$  $\approx$  0.9515396564 + 0.5263246529i
$L(1,\chi)$  $\approx$  0.9515396564 + 0.5263246529i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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