L-function 1-85-85.22-r0-0-0

• Label: 1-85-85.22-r0-0-0
• Degree: $1$
• Conductor: $85$
• Sign: $0.922 - 0.386i$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$85$    =    $5 \cdot 17$
Sign:	$0.922 - 0.386i$
Analytic conductor:	$0.394738$
Root analytic conductor:	$0.394738$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{85} (22, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 85,\ (0:\ ),\ 0.922 - 0.386i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9153440286 - 0.1838165876i$
$L(1)$	$\approx$	$0.9629729231 - 0.1667101039i$

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

−31.02270612493989696073537259282, −29.659874032527615353840679648497, −28.57629545318949943937993959361, −27.43145103862959100583319081446, −26.36871693050308580090879617432, −25.51365523857429334595703506864, −24.71666896192574438406384021808, −23.87218516935757756989925194731, −22.39164009731681268498756881142, −20.88136723636168541699612804860, −19.688716625135326199027415533566, −18.72073543975814796777023019145, −18.15649067663381311789609013316, −16.40398747024723385515677199199, −15.56861997693161397342861815166, −14.40105141544392212802098517723, −13.459502349675083138737016394737, −11.834609752335949660615392483294, −10.11731908490325898745576309317, −8.85468922881508184730955885268, −8.30570068255287426157807671635, −6.76917640314439923140342171327, −5.69419081218228498788741865719, −3.42053129634810597613967514474, −1.63821709722534523897851063561, 1.691482926362417444885704236761, 3.33643157289803732815401311268, 4.291399181366467096049173106058, 7.0084981767532850879113275372, 8.072908086531523831979301224615, 9.40633484419589703967285901171, 10.1143483925876720813396415097, 11.405703658796288540761325182, 13.0225402860067333200088001937, 13.83834688551077615783614421403, 15.48508552928735558890126003430, 16.55188619366019991226596211392, 17.802871817731663455393288912631, 19.03347652624144040646956144635, 20.14809787698167031356364142734, 20.48507812825529800837355366159, 21.8168896187675971020326870154, 22.94879109904264248986169386884, 24.69443015540158179839122647923, 26.072198900305498811638704289346, 26.17013534316152355455189828806, 27.589299056977549218198934356702, 28.33808722609564366757171051358, 29.87211484345893164568997686294, 30.46086225467897291674456773004

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