Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $-0.109 - 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  i·2-s + 3-s − 4-s i·6-s + 7-s + i·8-s + 9-s i·11-s − 12-s i·13-s i·14-s + 16-s i·18-s + 19-s + 21-s − 22-s + ⋯
L(s,χ)  = 1  i·2-s + 3-s − 4-s i·6-s + 7-s + i·8-s + 9-s i·11-s − 12-s i·13-s i·14-s + 16-s i·18-s + 19-s + 21-s − 22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.109 - 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.109 - 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $-0.109 - 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (47, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ -0.109 - 0.994i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.552661560 - 1.732427409i$
$L(\frac12,\chi)$  $\approx$  $1.552661560 - 1.732427409i$
$L(\chi,1)$  $\approx$  1.280511764 - 0.8261655281i
$L(1,\chi)$  $\approx$  1.280511764 - 0.8261655281i

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

−31.10165256550767751335865771973, −30.219028757455193982071587274057, −28.28877940412033120418798060713, −27.26580858313880318351309456040, −26.32691247833549452842053799059, −25.524378011040799976324987832, −24.4121846718429285823387925658, −23.80553442031721015080153133895, −22.2753198339994617903299746002, −21.117205427773709251693657394644, −19.99962026473324583163539164302, −18.5832965845912108990197495173, −17.79268643485250911168033825395, −16.37080730771282145380492382137, −15.19149240079811775147974278314, −14.38998405906814853255589924472, −13.554501827623410741556858237116, −12.05987586363985860154755497514, −10.010817843026440351879006374973, −8.948330872556021501096954856943, −7.842274551943836783887965299907, −6.93863189475755872031893417487, −5.0301142574548057662150230069, −3.92784700689315787262061837594, −1.7796478929075105273809192022, 1.176106853049164131082101619958, 2.67418230872795891753279086185, 3.86956913939863321187231323650, 5.35548632544261363862956244815, 7.83622832974468421772966160594, 8.584946354600817400219905585867, 9.92103509076293537279364771249, 11.05944395348454049385952772907, 12.34928711571515755526733759065, 13.69700981263997455540675680727, 14.29365613541970192209324307922, 15.706536224782504511878204855703, 17.582837755424516229660120260083, 18.50444644712295458374048674126, 19.59981074255785622594569064337, 20.52493551116370326675774938024, 21.288063807860990291137562085528, 22.35081382329642929394100652564, 23.91505715474301855040720936170, 24.82871291814530971702850907667, 26.359988200715468281610971869738, 27.0919152680202705147148463556, 27.976830804091730242585522989792, 29.47005473174143308108260159039, 30.29486194121357449613716011506

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