L-function 1-1020-1020.83-r1-0-0

• Label: 1-1020-1020.83-r1-0-0
• Degree: $1$
• Conductor: $1020$
• Sign: $-0.808 + 0.588i$
• Analytic cond.: $109.614$
• Root an. cond.: $109.614$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.707 + 0.707i)7^-s + (−0.707 − 0.707i)11^-s − i·13^-s − i·19^-s + (0.707 − 0.707i)23^-s + (0.707 − 0.707i)29^-s + (0.707 − 0.707i)31^-s + (0.707 + 0.707i)37^-s + (−0.707 − 0.707i)41^-s − 43^-s − i·47^-s − i·49^-s + 53^-s − i·59^-s + (0.707 + 0.707i)61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign:	$-0.808 + 0.588i$
Analytic conductor:	\(109.614\)
Root analytic conductor:	\(109.614\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1020} (83, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1020,\ (1:\ ),\ -0.808 + 0.588i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1213690909 + 0.3727650474i\)
\(L(1)\)	\(\approx\)	\(0.8424440518 + 0.01572271920i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	5	\( 1 \)
	17	\( 1 \)
good	7	\( 1 + (-0.707 + 0.707i)T \)
	11	\( 1 + (-0.707 - 0.707i)T \)
	13	\( 1 - iT \)
	19	\( 1 - iT \)
	23	\( 1 + (0.707 - 0.707i)T \)
	29	\( 1 + (0.707 - 0.707i)T \)
	31	\( 1 + (0.707 - 0.707i)T \)
	37	\( 1 + (0.707 + 0.707i)T \)
	41	\( 1 + (-0.707 - 0.707i)T \)

Imaginary part of the first few zeros on the critical line

−21.26804829796028805358381887953, −20.07755872478067942224987921443, −19.7347592681623675394568616757, −18.79108607855333107770397182486, −17.95370272803496858343480570074, −17.1566178302642190502062632513, −16.38308371972801562777780413145, −15.66236336193098433053214331162, −14.85285706323602060489084996054, −13.78784069300649935293995259275, −13.26639923173342010420392601924, −12.427288870348768925577405805316, −11.477120583103153319793000938448, −10.60992628333848760359498990118, −9.82064797748623152881339200447, −9.11450307140969020715661965703, −8.04688057322745023895458031918, −6.91212941213540950441213730179, −6.730108484563433053176174196587, −5.2324090321113664270494472271, −4.52679772931712002966359272704, −3.47156200685056773194082244721, −2.54148401237488508349499902631, −1.3387911514533699525985371546, −0.09350330028341258628129553101, 0.998435493318008408305499064850, 2.58203777445188467790941796530, 3.03642993005948384837304326258, 4.24360297174757379877163891738, 5.487118733988685671456115847722, 5.94284270455655441090086934072, 6.97520095120514544614352686012, 8.1914555025917565278133814874, 8.55284436658780926860507366105, 9.87083471037114225275913874649, 10.300235384314974825442891068351, 11.40380426102483214632519070701, 12.249341211293632823581008151723, 13.02435446703305865493653712000, 13.626210628613516451248051822858, 14.822862537118956162313409798170, 15.42957964269992642151301883752, 16.19668567034045374072809252235, 16.922348950034628618949471973822, 17.949016772465186084202978672415, 18.74720563368563607032046006015, 19.15026709439314433456925077777, 20.25161570827456851345515508530, 20.94861275826634462722403553075, 21.7196152134793449744089432863

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