L-function 1-3040-3040.1333-r0-0-0

• Label: 1-3040-3040.1333-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $0.712 + 0.701i$
• Analytic cond.: $14.1177$
• Root an. cond.: $14.1177$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.996 − 0.0871i)3^-s + (−0.5 + 0.866i)7^-s + (0.984 + 0.173i)9^-s + (0.258 − 0.965i)11^-s + (0.0871 + 0.996i)13^-s + (−0.984 + 0.173i)17^-s + (0.573 − 0.819i)21^-s + (−0.939 + 0.342i)23^-s + (−0.965 − 0.258i)27^-s + (−0.573 − 0.819i)29^-s + (0.5 − 0.866i)31^-s + (−0.342 + 0.939i)33^-s + (0.707 − 0.707i)37^-s − i·39^-s + (0.642 + 0.766i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$0.712 + 0.701i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1333, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ 0.712 + 0.701i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8019078232 + 0.3286405096i$
$L(1)$	$\approx$	$0.7172426493 + 0.06546141251i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (-0.996 - 0.0871i)T$
	7	$1 + (-0.5 + 0.866i)T$
	11	$1 + (0.258 - 0.965i)T$
	13	$1 + (0.0871 + 0.996i)T$
	17	$1 + (-0.984 + 0.173i)T$
	23	$1 + (-0.939 + 0.342i)T$
	29	$1 + (-0.573 - 0.819i)T$
	31	$1 + (0.5 - 0.866i)T$
	37	$1 + (0.707 - 0.707i)T$

Imaginary part of the first few zeros on the critical line

−18.83916332115951005300655970721, −17.94103824169205768708674528624, −17.6132057018519221645643541706, −16.979675349147382090344101660865, −16.10920623162121035037901177429, −15.71542127817008691607980837536, −14.874532597440910967570613223048, −13.97991761051898578454453854467, −13.088548387054800013309896837432, −12.64147244047144514018218052365, −11.95654812523023674863065491763, −10.99413670031800602495847390725, −10.50914626931918617768571775595, −9.886005472864625543680795294622, −9.170073443147638476070735243683, −7.993288035664148617412898360182, −7.2132271307318615936538943870, −6.66963638343243984539492457985, −5.94013317955436347290279889569, −5.02210913570222184850454128163, −4.327012617531709775018154421762, −3.6837091140633050336611438145, −2.510556906867547447288985823047, −1.41553592686261071997502729168, −0.47896310523551733354843914394, 0.67981862214822457512872837287, 1.88715876917547867956759731631, 2.59611356304946708249977860183, 3.93893248730855583264314851439, 4.35017064658514129227157889598, 5.6007827286828829819371099108, 6.02197428042831598763652287167, 6.54667177250278443082353008210, 7.505650385903674646518444976612, 8.4247856639929222823158652658, 9.31689260334261236961286683779, 9.726628228977510725069757315770, 10.94214817643980718614810445815, 11.31787494934309464149882923313, 11.983298625493657518251815486831, 12.671640593249457708185759534345, 13.443199022173632809047216153907, 14.06705672769856818068863857999, 15.20327138893688017807460470710, 15.75782140566339180628353946169, 16.37567618466650380020829588949, 16.95493720821749687304578977794, 17.76072904701519516997559263135, 18.42741811932857958808025139254, 19.07525652795466528986282706547

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