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Properties

Label 1-85-85.84-r0-0-0
Degree $1$
Conductor $85$
Sign $1$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85} (84, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9193904059\)
\(L(\frac12)\) \(\approx\) \(0.9193904059\)
\(L(1)\) \(\approx\) \(0.9585496198\)
\(L(1)\) \(\approx\) \(0.9585496198\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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