L-function 1-1020-1020.47-r1-0-0

• Label: 1-1020-1020.47-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:	$\chi_{1020} (47, \cdot )$
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

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

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