L-function 1-2652-2652.107-r1-0-0

• Label: 1-2652-2652.107-r1-0-0
• Degree: $1$
• Conductor: $2652$
• Sign: $0.157 + 0.987i$
• Analytic cond.: $284.996$
• Root an. cond.: $284.996$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.923 + 0.382i)5^-s + (−0.793 − 0.608i)7^-s + (−0.991 − 0.130i)11^-s + (−0.965 − 0.258i)19^-s + (0.991 + 0.130i)23^-s + (0.707 + 0.707i)25^-s + (0.793 − 0.608i)29^-s + (−0.382 + 0.923i)31^-s + (−0.5 − 0.866i)35^-s + (−0.608 − 0.793i)37^-s + (−0.130 + 0.991i)41^-s + (0.258 − 0.965i)43^-s − i·47^-s + (0.258 + 0.965i)49^-s + (0.707 − 0.707i)53^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2652$    =    $2^{2} \cdot 3 \cdot 13 \cdot 17$
Sign:	$0.157 + 0.987i$
Analytic conductor:	$284.996$
Root analytic conductor:	$284.996$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2652} (107, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ 0.157 + 0.987i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9799122441 + 0.8360376421i$
$L(1)$	$\approx$	$0.9910196606 + 0.02224219668i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	13	$1$
	17	$1$
good	5	$1 + (0.923 + 0.382i)T$
	7	$1 + (-0.793 - 0.608i)T$
	11	$1 + (-0.991 - 0.130i)T$
	19	$1 + (-0.965 - 0.258i)T$
	23	$1 + (0.991 + 0.130i)T$
	29	$1 + (0.793 - 0.608i)T$
	31	$1 + (-0.382 + 0.923i)T$
	37	$1 + (-0.608 - 0.793i)T$
	41	$1 + (-0.130 + 0.991i)T$

Imaginary part of the first few zeros on the critical line

−18.84150672070764910797691695292, −18.36781413266783033124188585962, −17.597353833355692004831409809719, −16.77907895650265358520184151186, −16.32876152598601683447318867707, −15.34784089432228849459325386618, −14.92614268217798585159571212591, −13.78539873551663228798601156793, −13.26190861142683009261241496632, −12.62319993879396398294572741042, −12.10760103900973250433556724647, −10.84275607178794925745945204741, −10.31334755482153143688114197483, −9.57893182565259200884509708475, −8.85166335210955379933740642498, −8.285683290466740306487955590409, −7.11389362792809161390494995193, −6.41894785870351575443077023310, −5.62381980284666744066393885484, −5.10190578760549968443592211029, −4.10300427088721723589485770779, −2.879982383193453040671282041233, −2.44192834190759366866583938744, −1.42134752442667072913226313407, −0.25535844770289608711438280624, 0.73477510693113019208574361991, 1.88204027081675604460327164255, 2.762923194653584895240671436, 3.35463728518976891977413298512, 4.497076498176006038278767535782, 5.3101403739756620952277123320, 6.13780125112278629006668302615, 6.79436040357827377240674309172, 7.44230440969674882952015638725, 8.50912230830208484822836735114, 9.24953422898702365790432056934, 10.068940093756944904284167670480, 10.57059397377502004729436857511, 11.12313982553246067984723760478, 12.42184789563632189124521136166, 13.01969658043113426236011986329, 13.522544427742273808018776899330, 14.233556759411324804574875734252, 15.04007008386791274490902577719, 15.84534329536755317686679545418, 16.48612830330334186627769882027, 17.36785922057374191395419488747, 17.70365111781272853619001625288, 18.73617778898962956914204545477, 19.16261826999029050450236500543

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