Properties

Label 1-85-85.74-r1-0-0
Degree $1$
Conductor $85$
Sign $-0.204 + 0.978i$
Analytic cond. $9.13451$
Root an. cond. $9.13451$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(9.13451\)
Root analytic conductor: \(9.13451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (1:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1878619976 + 0.2312853121i\)
\(L(\frac12)\) \(\approx\) \(0.1878619976 + 0.2312853121i\)
\(L(1)\) \(\approx\) \(0.4809171008 - 0.05021143568i\)
\(L(1)\) \(\approx\) \(0.4809171008 - 0.05021143568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (0.382 - 0.923i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 + T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.382 - 0.923i)T \)
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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

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

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