L-function 1-2793-2793.1871-r1-0-0

• Label: 1-2793-2793.1871-r1-0-0
• Degree: $1$
• Conductor: $2793$
• Sign: $0.948 + 0.316i$
• Analytic cond.: $300.149$
• Root an. cond.: $300.149$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.411 + 0.911i)2^-s + (−0.661 + 0.749i)4^-s + (0.998 − 0.0498i)5^-s + (−0.955 − 0.294i)8^-s + (0.456 + 0.889i)10^-s + (−0.0747 − 0.997i)11^-s + (0.270 − 0.962i)13^-s + (−0.124 − 0.992i)16^-s + (0.661 + 0.749i)17^-s + (−0.623 + 0.781i)20^-s + (0.878 − 0.478i)22^-s + (0.318 + 0.947i)23^-s + (0.995 − 0.0995i)25^-s + (0.988 − 0.149i)26^-s + (0.853 − 0.521i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2793$    =    $3 \cdot 7^{2} \cdot 19$
Sign:	$0.948 + 0.316i$
Analytic conductor:	$300.149$
Root analytic conductor:	$300.149$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2793} (1871, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2793,\ (1:\ ),\ 0.948 + 0.316i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.311793504 + 0.5380728097i$
$L(1)$	$\approx$	$1.418573130 + 0.5431431342i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	19	$1$
good	2	$1 + (0.411 + 0.911i)T$
	5	$1 + (0.998 - 0.0498i)T$
	11	$1 + (-0.0747 - 0.997i)T$
	13	$1 + (0.270 - 0.962i)T$
	17	$1 + (0.661 + 0.749i)T$
	23	$1 + (0.318 + 0.947i)T$
	29	$1 + (0.853 - 0.521i)T$
	31	$1 + T$
	37	$1 + (-0.988 - 0.149i)T$

Imaginary part of the first few zeros on the critical line

−18.83207603879305656981965619392, −18.54112102045395212504896361118, −17.73689987443436520566094063363, −17.11197711347534553244762357605, −16.22217404996306380341414052030, −15.252201126419873954293509266222, −14.45425959127976979257583012786, −13.96109517647202323273863620694, −13.39475677100926585488432302229, −12.44674614884793351752319103444, −12.077833057092013788820209108203, −11.1503175233170804926283565667, −10.24820355240728339899251997987, −9.942275590390756558465542084198, −9.09744526169621211817062941519, −8.512519491447206296390868444015, −7.08702534573933742306048995806, −6.51324809991859839471494274257, −5.57279451803209872416763043745, −4.8421036525236903701203243721, −4.266273588384164305251338495217, −3.08258575723678992834273182095, −2.43002796628505523588601218130, −1.64684808685285773711582341463, −0.88392915645759436412436188733, 0.52991904228297840670164164513, 1.486759454636116269993019372040, 2.925859499826986324545367124308, 3.29830848055294997818658273177, 4.47669841255995585961386493731, 5.330506084238398842810968151966, 5.94251860145764805147100576988, 6.331420232053596785745253844935, 7.43194275598035907952327321457, 8.21287672332516815587582501991, 8.72718732408769869634130079500, 9.6745653732693572427635396267, 10.311535204860317483700761207365, 11.23260521074238909675725135698, 12.263046880749544691333272189805, 12.92782427859681822825990125500, 13.59317607789817242102055457349, 14.024765212345246369735648751200, 14.836298759448883048863447084, 15.578402150643323359789168704243, 16.21477996320251364919859589021, 17.04309893062589020583547263980, 17.48574494163093723002454133236, 18.101864711364614307049291486111, 18.92494792366274226765302543536

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