L-function 1-2793-2793.269-r1-0-0

• Label: 1-2793-2793.269-r1-0-0
• Degree: $1$
• Conductor: $2793$
• Sign: $-0.609 + 0.792i$
• 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

+ (0.698 + 0.715i)2^-s + (−0.0249 + 0.999i)4^-s + (−0.797 − 0.603i)5^-s + (−0.733 + 0.680i)8^-s + (−0.124 − 0.992i)10^-s + (−0.826 + 0.563i)11^-s + (−0.411 − 0.911i)13^-s + (−0.998 − 0.0498i)16^-s + (−0.0249 − 0.999i)17^-s + (0.623 − 0.781i)20^-s + (−0.980 − 0.198i)22^-s + (−0.878 + 0.478i)23^-s + (0.270 + 0.962i)25^-s + (0.365 − 0.930i)26^-s + (−0.661 − 0.749i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2793$    =    $3 \cdot 7^{2} \cdot 19$
Sign:	$-0.609 + 0.792i$
Analytic conductor:	$300.149$
Root analytic conductor:	$300.149$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2793} (269, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2793,\ (1:\ ),\ -0.609 + 0.792i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3919027174 + 0.7962483159i$
$L(1)$	$\approx$	$0.9576715986 + 0.3118074580i$

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	$1 + (0.698 + 0.715i)T$
	5	$1 + (-0.797 - 0.603i)T$
	11	$1 + (-0.826 + 0.563i)T$
	13	$1 + (-0.411 - 0.911i)T$
	17	$1 + (-0.0249 - 0.999i)T$
	23	$1 + (-0.878 + 0.478i)T$
	29	$1 + (-0.661 - 0.749i)T$
	31	$1 + T$
	37	$1 + (-0.365 - 0.930i)T$

Imaginary part of the first few zeros on the critical line

−19.04334825371103919327224726130, −18.421972441958739247062197679185, −17.57945725545567921955685334088, −16.40778088895435493251641516513, −15.8375089546724964441505955777, −15.13414594745393559568036865639, −14.38233749037953151286970939921, −13.97674064639882519262806574444, −12.93558205927328434583331888337, −12.4219136117648127539351889413, −11.57507959020928167946578238452, −11.08767262956295828189975652682, −10.38204772566571404855082041529, −9.74839632245282815366247599396, −8.63382884556948299148933394779, −7.95209470843418096186334784273, −6.92627213199225228599724701766, −6.28089039398505633929893536671, −5.44232888630684737970443758870, −4.43898913370884923520612525814, −3.95402248564641739972120771566, −3.02618986984849759116615168178, −2.40968794099538150119575595182, −1.38721756787496317837220556683, −0.17688313209830646630163667393, 0.56422197647986622662392205030, 2.17324331261570390266725947556, 2.98423282706431328115280001835, 3.88128186142572547187000898771, 4.623311244549181967645744626179, 5.25629775443684493469813940376, 5.89790226174787953360642680899, 7.09219222731913348706244536840, 7.67816902589039751436990238196, 8.05317887594199829599192692562, 9.032008167891345226614905420563, 9.83776479792218341140472253882, 10.86949971130740828400782282765, 11.83600891171992271420880058297, 12.21007838164787319043948264465, 13.023760564326874785147392882358, 13.51658720911325318104286651501, 14.4606019922036710017769611144, 15.24204792931347526000369492987, 15.74325282022839505086827983697, 16.15744724266755292766525347165, 17.12152990326139163778625638664, 17.687258866877863705193104133739, 18.40413226795666300020613031101, 19.396814171505561749214917940029

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