L-function 1-1020-1020.383-r1-0-0

• Label: 1-1020-1020.383-r1-0-0
• Degree: $1$
• Conductor: $1020$
• Sign: $-0.961 - 0.274i$
• 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign:	$-0.961 - 0.274i$
Analytic conductor:	\(109.614\)
Root analytic conductor:	\(109.614\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1020} (383, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1020,\ (1:\ ),\ -0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1702417816 - 1.215294868i\)
\(L(1)\)	\(\approx\)	\(0.9095438813 - 0.3188006766i\)

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

−21.80843165931536623925284823805, −21.056045473912147857871114172815, −20.13302871497215614551561471113, −19.40768887230283172258265667580, −18.630879047972258108184530311614, −18.065187393992239200963590193580, −16.70470717816703083763653216700, −16.576509277614140515512059974621, −15.400384654495562238138066402846, −14.68744977032411643121660042911, −14.00489276638420738845155635894, −12.77391791100402636993225795281, −12.36971046678304891678379344028, −11.49203501603969168225118006232, −10.53353117922269875646092242587, −9.38589788367270246706569747382, −9.18189068982413511117337554715, −7.99471860124877981516244857618, −6.87547427690869893760241820812, −6.35259262876205418664395932273, −5.290725196930856228461448200680, −4.27924087308245960013585326133, −3.37846765247396179739471078354, −2.27295951150697397593290266416, −1.3390374681719929022423092083, 0.28776574445013575804828130917, 1.05789884453765646357483985284, 2.58019520627107331942401836905, 3.474787944287541350229671016713, 4.235946098646904707219587513, 5.50368731220170393157970895065, 6.23013637289188869579609211928, 7.22536651162779614530319732016, 7.89598663056505681781630516279, 9.18306142237124269081607946339, 9.56660484144410624692658712633, 10.88679032417878828439451607568, 11.15097155020398324514815530490, 12.47829741832790813415606841980, 13.15504954659929499004602417165, 13.8029179993807399555340790541, 14.76132178641447363582716479160, 15.61708121660793270896215202832, 16.35595459933799240621189479086, 17.21035393550642281164628739800, 17.71333559417421824737148184057, 18.98092710080265676476955365542, 19.4476338644929036858816731386, 20.21328729655919378539026443453, 20.96690585699762356505956399482

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