L-function 1-1020-1020.803-r1-0-0

• Label: 1-1020-1020.803-r1-0-0
• Degree: $1$
• Conductor: $1020$
• Sign: $0.109 - 0.994i$
• 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

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign:	$0.109 - 0.994i$
Analytic conductor:	\(109.614\)
Root analytic conductor:	\(109.614\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1020} (803, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1020,\ (1:\ ),\ 0.109 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.3985999272 - 0.3572390865i\)
\(L(1)\)	\(\approx\)	\(0.7822378056 + 0.08586798725i\)

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	\( 1 \)
	11	\( 1 \)
	13	\( 1 \)
	19	\( 1 - T \)
	23	\( 1 \)
	29	\( 1 \)
	31	\( 1 \)
	37	\( 1 + iT \)
	41	\( 1 \)

Imaginary part of the first few zeros on the critical line

−21.78375201926122403950032332328, −20.75331161053254686939704882825, −19.96805531210236145447724018364, −19.2445406648592261133168999574, −18.614770907344032810979598680078, −17.66910989347776786085224914873, −16.800741127619450986492329081268, −16.10331587609503258963139543525, −15.42815920652089975080181913589, −14.48502718941012021143087963526, −13.55666427667315144640169939918, −12.89757247391767430983688795224, −12.18814556294934508647597122673, −11.06991286749198588605063621238, −10.38283993159428500043871645931, −9.56589308948109851725725793858, −8.59403780538592577519365221712, −7.90453687758854747059201584, −6.715593752228408560257054182665, −6.049771133291691914059914221699, −5.22075783408000687554784867752, −3.87819282721481884503934408959, −3.22934054931733469915164403883, −2.20285123130445775250619826738, −0.72826405751635583084537900878, 0.149067710807986345437112426, 1.72111604475715168029474814125, 2.54212182436909366712081808375, 3.81270746642929449185163378360, 4.43509333191583258160065843823, 5.6607159883793694383662139113, 6.61600490572977640881383631627, 7.12541972791402199400224029681, 8.329999287141880845255779661557, 9.211677511724328941039113097477, 9.947913042315806524927548660148, 10.64051811434987107276228059997, 11.90747302998787354967284661648, 12.368954136867811632327528906346, 13.301220168946344076349073447002, 14.06056200547392031322131049292, 14.998513909817693469414496088728, 15.76432743745914410877847190161, 16.4942223636273387626055232568, 17.28781050058118426354189900139, 18.09285354699277914524879569581, 19.10365616502001745385002958421, 19.51206317596367637755927226703, 20.42890354171744661978401835259, 21.26673761872062208634851722029

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