Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $0.485 + 0.874i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s i·13-s + (−0.707 − 0.707i)14-s + 16-s + i·18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯
L(s,χ)  = 1  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s − 8-s i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s i·13-s + (−0.707 − 0.707i)14-s + 16-s + i·18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.485 + 0.874i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.485 + 0.874i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $0.485 + 0.874i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (43, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ 0.485 + 0.874i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7833983283 + 0.4609829046i$
$L(\frac12,\chi)$  $\approx$  $0.7833983283 + 0.4609829046i$
$L(\chi,1)$  $\approx$  0.6552752360 + 0.1872600781i
$L(1,\chi)$  $\approx$  0.6552752360 + 0.1872600781i

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

−30.25677765074906886998491640916, −29.04609075099328075872996652390, −28.256246629327622456307699586154, −27.28983070683064921500005966461, −26.21090642838581081596960572018, −24.922292908204866899735347782387, −24.126078078360455828348576511477, −23.13260363322880235981397961304, −21.60900884543719912345772009101, −20.29450250841027670367286308510, −19.29346997590011269633793210145, −18.227073448434889826842660019920, −17.22436845361609205357524964232, −16.712875157987187688728003316291, −15.071306401302339897487255872790, −13.61021681615192295928604314233, −11.98260000413246335450481872326, −11.29150273280080257914189834735, −10.06755472260077860976640012735, −8.5538608238709414943197369519, −7.23186575617702492602606225595, −6.53258958451520037386644414331, −4.63520506642173997669643698013, −2.0927668191575769470539189485, −0.82207299266547469513904884958, 1.1383928669078461898728057337, 3.25191008899379840390193298697, 5.27725480453999825071742208815, 6.35856799748482159972090840437, 8.111924088683160970153428614145, 9.17588841084002884507208584087, 10.41413955352695852481313430498, 11.381565052584968141483091721392, 12.28767368057502739504808960030, 14.62150484261558793713302513992, 15.5647198294258681607023919429, 16.62827935005594759739419676076, 17.597092138614138189897198911185, 18.45285045593323549063307244188, 19.83307618955628609864527583326, 21.06546068743431416671799833025, 21.79111642061991787778286406045, 23.22841099127827577468152893903, 24.582080390254377036312065152416, 25.39373991314070522331572407692, 27.12255615421328640770464424429, 27.23403797606684176405914985635, 28.3421429976376531877305404041, 29.30179402333211565732270344877, 30.314031795164744739008107194995

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