Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $-0.204 - 0.978i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.707 + 0.707i)2-s + (−0.923 − 0.382i)3-s i·4-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)12-s i·13-s + (−0.923 − 0.382i)14-s − 16-s − 18-s + (0.707 − 0.707i)19-s i·21-s + (0.382 − 0.923i)22-s + ⋯
L(s,χ)  = 1  + (−0.707 + 0.707i)2-s + (−0.923 − 0.382i)3-s i·4-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)12-s i·13-s + (−0.923 − 0.382i)14-s − 16-s − 18-s + (0.707 − 0.707i)19-s i·21-s + (0.382 − 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.204 - 0.978i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.204 - 0.978i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $-0.204 - 0.978i$
motivic weight  =  \(0\)
character  :  $\chi_{85} (54, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (1:\ ),\ -0.204 - 0.978i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1878619976 - 0.2312853121i$
$L(\frac12,\chi)$  $\approx$  $0.1878619976 - 0.2312853121i$
$L(\chi,1)$  $\approx$  0.4809171008 + 0.05021143568i
$L(1,\chi)$  $\approx$  0.4809171008 + 0.05021143568i

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

−30.482180397220817326528941165929, −29.317735470929441484965305746465, −28.84395667719613496436849453171, −27.64028340433329489059356372367, −26.78559616542454034515236261653, −26.07460142779635962469057526176, −24.20956599625913887453424478791, −23.23018453874684405125348735769, −21.95716935058511835136291219121, −21.07894674627741285297006271362, −20.15950539482900242082183281171, −18.63054839976274887036657843304, −17.856404308542264686669981956605, −16.654243014643153951838501866384, −16.07837546363239261166771091056, −14.03253496192393207100377111714, −12.62305802356221352749990681248, −11.45802778738970724609660217015, −10.614690702425773311077217668629, −9.68741508385222667054882592699, −8.06234371271459680084749926930, −6.76737320987193677819206777395, −4.88455369938022644989795511810, −3.5938090885561213342213968469, −1.42138943384974648112081806380, 0.20717594285348801576894711785, 2.05535783971545314584385748347, 5.10405808486258639897821286812, 5.75859644771345418414426010573, 7.25919075136511141366955511074, 8.24650388555790742650528458539, 9.84203684821214663939938273102, 10.944079130629956101932451759725, 12.18755152687751658093604978137, 13.5819883420827389088827879430, 15.307854654603355406482362014721, 15.88538543618321303094180207356, 17.38456534186120695987301505320, 18.03516476421401433303106259211, 18.82266520730370156309829561181, 20.27894555354672375915318399870, 21.892107692100140916204205171921, 22.97712378417040217263076579158, 24.00408947609661202276198087420, 24.79742907487160203758010516383, 25.844016246854900445272981748325, 27.21701073628826965412565761872, 28.14290751331128858501107776558, 28.69252135137716775852499350623, 29.96083612625974683770547708729

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