L-function 1-85-85.73-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	17	$1$
good	2	$1 + (0.707 + 0.707i)T$
	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	$1 + (0.382 - 0.923i)T$

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