Properties

Degree 1
Conductor $ 5 \cdot 17 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 13-s − 14-s + 16-s − 18-s + 19-s + 21-s + 22-s + 23-s − 24-s + 26-s + 27-s + 28-s − 29-s − 31-s − 32-s − 33-s + 36-s + 37-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 13-s − 14-s + 16-s − 18-s + 19-s + 21-s + 22-s + 23-s − 24-s + 26-s + 27-s + 28-s − 29-s − 31-s − 32-s − 33-s + 36-s + 37-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(85\)    =    \(5 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{85} (84, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 85,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9193904059$
$L(\frac12,\chi)$  $\approx$  $0.9193904059$
$L(\chi,1)$  $\approx$  0.9585496198
$L(1,\chi)$  $\approx$  0.9585496198

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

−30.72448238809539594394871249403, −29.65645835098769905229593805733, −28.55521132706340923944105584234, −27.15564406817380866968360307697, −26.75664272122655833896616298635, −25.59305377062313480100512656699, −24.60851027894432967892895400801, −23.87126753796721754580676058823, −21.68946210944048905891715962851, −20.71877589620573646711382075406, −20.00168473732113438799068479433, −18.74053864541238034785930501280, −17.99460262896709528946428395818, −16.6377597229822915235116626299, −15.309167343015127879565125059008, −14.560405591742370004622066385005, −12.98850524492139875851288847610, −11.48298487623341184948500281464, −10.21053358708734682080616972270, −9.13309214990599921422387119046, −7.93023789479072131087529224943, −7.278194548717802760674116138725, −5.097566354913826213861931440856, −3.013620999828644591667127063911, −1.75250756366728686044303364177, 1.75250756366728686044303364177, 3.013620999828644591667127063911, 5.097566354913826213861931440856, 7.278194548717802760674116138725, 7.93023789479072131087529224943, 9.13309214990599921422387119046, 10.21053358708734682080616972270, 11.48298487623341184948500281464, 12.98850524492139875851288847610, 14.560405591742370004622066385005, 15.309167343015127879565125059008, 16.6377597229822915235116626299, 17.99460262896709528946428395818, 18.74053864541238034785930501280, 20.00168473732113438799068479433, 20.71877589620573646711382075406, 21.68946210944048905891715962851, 23.87126753796721754580676058823, 24.60851027894432967892895400801, 25.59305377062313480100512656699, 26.75664272122655833896616298635, 27.15564406817380866968360307697, 28.55521132706340923944105584234, 29.65645835098769905229593805733, 30.72448238809539594394871249403

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