L-function 1-812-812.347-r1-0-0

• Label: 1-812-812.347-r1-0-0
• Degree: $1$
• Conductor: $812$
• Sign: $0.0633 + 0.997i$
• Analytic cond.: $87.2615$
• Root an. cond.: $87.2615$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 − 0.866i)3^-s + (−0.5 + 0.866i)5^-s + (−0.5 + 0.866i)9^-s + (−0.5 − 0.866i)11^-s + 13^-s + 15^-s + (0.5 + 0.866i)17^-s + (−0.5 + 0.866i)19^-s + (0.5 − 0.866i)23^-s + (−0.5 − 0.866i)25^-s + 27^-s + (−0.5 − 0.866i)31^-s + (−0.5 + 0.866i)33^-s + (0.5 − 0.866i)37^-s + (−0.5 − 0.866i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$812$    =    $2^{2} \cdot 7 \cdot 29$
Sign:	$0.0633 + 0.997i$
Analytic conductor:	$87.2615$
Root analytic conductor:	$87.2615$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{812} (347, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 812,\ (1:\ ),\ 0.0633 + 0.997i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5614307232 + 0.5269315854i$
$L(1)$	$\approx$	$0.7622891505 - 0.04837501029i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	7	$1$
	29	$1$
good	3	$1 + (-0.5 - 0.866i)T$
	5	$1 + (-0.5 + 0.866i)T$
	11	$1 + (-0.5 - 0.866i)T$
	13	$1 + T$
	17	$1 + (0.5 + 0.866i)T$
	19	$1 + (-0.5 + 0.866i)T$
	23	$1 + (0.5 - 0.866i)T$
	31	$1 + (-0.5 - 0.866i)T$
	37	$1 + (0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−21.7267848234480448493959290193, −20.92917918928380709349241018532, −20.46385658263112819996434619679, −19.68546646122173039980168173429, −18.50780150007973560439502629491, −17.65285249572490961715142807781, −16.92307197783916533730766831446, −16.053769579142177118419468870518, −15.595280353514834723129808807351, −14.857804729738535901168565535761, −13.57934277252861401849496782550, −12.76978498163492718902711956187, −11.85962970111694143983973585703, −11.21935508024867871057625449605, −10.28748889191084313145769100451, −9.35012802136372246049413529517, −8.74202434043651251302462235256, −7.66032280070924937583572514383, −6.618144552882920705372503587326, −5.35836300048889519827661101812, −4.88831328511795127593378235429, −3.96939109969750602074579080470, −3.01292502119438915290991084737, −1.369325513078973092591180748131, −0.23672897035108542670976918796, 0.901741145461681564109977165088, 2.134796386401284095072344529669, 3.189873451847520553731392764676, 4.131591258507628024807962832607, 5.685721218087872046500027122864, 6.14501502186902630980790664087, 7.072119620461204177541936033200, 8.05567457432857238888659380189, 8.48710927969818409423688878459, 10.16444443363676246717531377312, 10.97165235507909395918590580471, 11.34265339761659150088475716043, 12.51811921834330966918660352797, 13.10198155884214055925903880554, 14.135087144184321118827765036335, 14.75974700068244382151211647568, 15.93546504068247136034561643774, 16.57052068018117053904820681492, 17.52200771247713770098369882006, 18.55938183299981002899191093007, 18.7564893664518066416978526123, 19.46858579700034480521072858080, 20.645110761429366718731919224653, 21.52203450487988214860795006572, 22.44318403196456501640993915798

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