L-function 1-2793-2793.1409-r0-0-0

• Label: 1-2793-2793.1409-r0-0-0
• Degree: $1$
• Conductor: $2793$
• Sign: $0.894 + 0.447i$
• Analytic cond.: $12.9706$
• Root an. cond.: $12.9706$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.980 + 0.198i)2^-s + (0.921 + 0.388i)4^-s + (0.583 + 0.811i)5^-s + (0.826 + 0.563i)8^-s + (0.411 + 0.911i)10^-s + (−0.623 − 0.781i)11^-s + (0.0249 − 0.999i)13^-s + (0.698 + 0.715i)16^-s + (0.124 − 0.992i)17^-s + (0.222 + 0.974i)20^-s + (−0.456 − 0.889i)22^-s + (0.797 − 0.603i)23^-s + (−0.318 + 0.947i)25^-s + (0.222 − 0.974i)26^-s + (−0.998 + 0.0498i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2793$    =    $3 \cdot 7^{2} \cdot 19$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$12.9706$
Root analytic conductor:	$12.9706$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2793} (1409, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2793,\ (0:\ ),\ 0.894 + 0.447i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.784662203 + 0.8935406993i$
$L(1)$	$\approx$	$2.158868952 + 0.4008227975i$

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	$1 + (0.980 + 0.198i)T$
	5	$1 + (0.583 + 0.811i)T$
	11	$1 + (-0.623 - 0.781i)T$
	13	$1 + (0.0249 - 0.999i)T$
	17	$1 + (0.124 - 0.992i)T$
	23	$1 + (0.797 - 0.603i)T$
	29	$1 + (-0.998 + 0.0498i)T$
	31	$1 + (0.5 + 0.866i)T$
	37	$1 + (0.733 - 0.680i)T$

Imaginary part of the first few zeros on the critical line

−19.27190235941229211928787321116, −18.67796662415720670743805025087, −17.5587974743141693503260822576, −16.89847497779788064049729092343, −16.393654683724317880291511756595, −15.35505538477847121041623176235, −15.02660161651133223057982244308, −14.02565063430833458086447154946, −13.38312144896350501002732213988, −12.90630995711144554874025102000, −12.20107879234470162550606322513, −11.52980121440351057590255138282, −10.634876132416536600543601058130, −9.90602970848861137538720783782, −9.24955938806863709246989922507, −8.26255678942122698418227922834, −7.33209588779239404699556748260, −6.62046868499142510328882331928, −5.656769421141746056084459386583, −5.24748638682387616268502592203, −4.28862927483708571476919934238, −3.81762006881446449175010690295, −2.43832174054993944814924656318, −1.976013235540507242767267077573, −1.05289589189023311218693296393, 0.97759322484489408510179169460, 2.32406651839747386641969409201, 2.89828302870251976241497805888, 3.420171016629541589803943637081, 4.61198412298529792867929135931, 5.44419794341392264726282548444, 5.8893929950640886362280919763, 6.76381901254827214792858531676, 7.45685929223134789031551084140, 8.1433501180370925183819269326, 9.22671245235877725153445487367, 10.21438965081257788927410158409, 10.88701497147790316092599831362, 11.288125400562323281724448340775, 12.360516425347251398399933997880, 13.08120092653554035474389183072, 13.610023291966038254934587595733, 14.291416208893483902436684995334, 14.937551193120355399146846175749, 15.58147344547891283880800376077, 16.331424582365381068391620424039, 17.00331331102734224215422636488, 17.9379207768167789835452060840, 18.45320700758079505551237715479, 19.32169925017922958096799477501

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