L-function 1-345-345.152-r1-0-0

• Label: 1-345-345.152-r1-0-0
• Degree: $1$
• Conductor: $345$
• Sign: $-0.247 + 0.968i$
• Analytic cond.: $37.0753$
• Root an. cond.: $37.0753$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.540 − 0.841i)2^-s + (−0.415 + 0.909i)4^-s + (−0.755 + 0.654i)7^-s + (0.989 − 0.142i)8^-s + (0.841 + 0.540i)11^-s + (0.755 + 0.654i)13^-s + (0.959 + 0.281i)14^-s + (−0.654 − 0.755i)16^-s + (−0.909 + 0.415i)17^-s + (0.415 − 0.909i)19^-s − i·22^-s + (0.142 − 0.989i)26^-s + (−0.281 − 0.959i)28^-s + (0.415 + 0.909i)29^-s + (−0.142 − 0.989i)31^-s + (−0.281 + 0.959i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$345$    =    $3 \cdot 5 \cdot 23$
Sign:	$-0.247 + 0.968i$
Analytic conductor:	$37.0753$
Root analytic conductor:	$37.0753$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{345} (152, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 345,\ (1:\ ),\ -0.247 + 0.968i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3921382068 + 0.5051420012i$
$L(1)$	$\approx$	$0.6933691942 - 0.05505722838i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
	23	$1$
good	2	$1 + (-0.540 - 0.841i)T$
	7	$1 + (-0.755 + 0.654i)T$
	11	$1 + (0.841 + 0.540i)T$
	13	$1 + (0.755 + 0.654i)T$
	17	$1 + (-0.909 + 0.415i)T$
	19	$1 + (0.415 - 0.909i)T$
	29	$1 + (0.415 + 0.909i)T$
	31	$1 + (-0.142 - 0.989i)T$
	37	$1 + (-0.281 + 0.959i)T$

Imaginary part of the first few zeros on the critical line

−24.78877685880224484961212372116, −23.489018419505529707563459979341, −22.87373799659994884668544672021, −22.11387989330017369127930978506, −20.60930537320835217011657097186, −19.737458007913672261274784982888, −19.05598583912888696204695731820, −17.9774118652524770879977483975, −17.24047297658216170922528238356, −16.17236914653822031171079423599, −15.82841887998998582598084224951, −14.45989299243961113138816982184, −13.76826922310699406816479117056, −12.85076902415903466910834885398, −11.330810543076762619299955527725, −10.40523019351099635690926430609, −9.48314944755137750258583077968, −8.58922097883806647416623247265, −7.55583253104889324632421265872, −6.518304227239364206216092803304, −5.87488798811633955758317078183, −4.43552434303631787137453312958, −3.32219604134544901146404320899, −1.377200882474929146470084528703, −0.242896423794674304146210842437, 1.35955897278689044480456495673, 2.49449359318241105312537500763, 3.62903084621176671711665223778, 4.64288879142506088584093222453, 6.284428730016917692816071714284, 7.18321208195235926660643316663, 8.74011345393212711330499653927, 9.11529338988321220843620453846, 10.145802348096123707762946149327, 11.29221766293685971790590760734, 11.98034045061920351607771203085, 12.96871069312916398760283439211, 13.73442069736950070331825166200, 15.14132141350871707586606295328, 16.12055594901854629672307878510, 17.03941097296215853870723328594, 17.993041549469445596388834989566, 18.760795799566645170603754391182, 19.66875804073503189678432784537, 20.23676043527631832952455952187, 21.449002745262696842412521901350, 22.10898229080896039663603029231, 22.78314996177385716511698534683, 24.05503455375019624225872604108, 25.24120207019421839634350151202

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