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Properties

Label 1-85-85.73-r0-0-0
Degree $1$
Conductor $85$
Sign $-0.0672 + 0.997i$
Analytic cond. $0.394738$
Root an. cond. $0.394738$
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 + (−0.382 − 0.923i)6-s + (0.382 + 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s − 13-s + (−0.382 + 0.923i)14-s − 16-s + i·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 + (−0.382 − 0.923i)6-s + (0.382 + 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s − 13-s + (−0.382 + 0.923i)14-s − 16-s + i·18-s + (0.707 − 0.707i)19-s − i·21-s + (−0.382 + 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0672 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0672 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.0672 + 0.997i$
Analytic conductor: \(0.394738\)
Root analytic conductor: \(0.394738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 85,\ (0:\ ),\ -0.0672 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7119721737 + 0.7615648685i\)
\(L(\frac12)\) \(\approx\) \(0.7119721737 + 0.7615648685i\)
\(L(1)\) \(\approx\) \(0.9642480719 + 0.5522556031i\)
\(L(1)\) \(\approx\) \(0.9642480719 + 0.5522556031i\)

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 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
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

−30.178862610654872686601371921288, −29.46377856434606464511205834934, −28.710794453366678051358885234000, −27.251236322120887640284788440686, −26.93926137913392384872603276865, −24.643478082461907982911136668197, −23.80227499896398337374821478247, −22.85591841401070606728388138511, −21.93482739481302151663191728202, −21.03630141462255951873818442052, −19.92867432883343144840675886522, −18.680317161523333984398565973656, −17.3176102748643916037113186894, −16.29435105821012980786809089578, −14.85693115485160348120801185696, −13.79990712914007177911427820132, −12.45829533695528217568972106529, −11.38908966035681649866506165795, −10.58218570984417844335183876668, −9.444366092782201870462512688189, −7.16143543030452948761413458359, −5.73922776767931742493230283191, −4.64754101297568639466517073421, −3.43638568954143145263750017115, −1.14813715685071839351923108299, 2.35910932539539867705045481879, 4.582173509587405477322745610515, 5.43525226383141049608089419618, 6.735741697945022185550354740678, 7.73585007335072522650885982882, 9.43030999403391821691639582545, 11.416382153818673334026647746128, 12.18952438993036098875810510849, 13.15151670670174937116614184195, 14.70896217923262512529824358033, 15.55930849271363417026429174147, 16.94924216649223516736493836343, 17.63007052328296346912229240006, 18.79765802226449380553908359628, 20.57515417809488288137136224889, 22.01594617359789964768590935272, 22.378420584732858494839078299150, 23.65257317164693537460960359605, 24.57250051362071088167273303208, 25.23556500071463016791703583696, 26.77967598218876911203321858596, 27.92822371867661842918456521327, 28.98701570593583853831902489542, 30.250565359797477304831752496243, 30.96179343326406070586417943195

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