L-function 1-3040-3040.117-r0-0-0

• Label: 1-3040-3040.117-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $0.309 - 0.951i$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$0.309 - 0.951i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (117, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ 0.309 - 0.951i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.372675182 - 1.723695317i$
$L(1)$	$\approx$	$1.637510885 - 0.4251473994i$

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

−19.215722539107101012641746609585, −18.581544131538365634788857739784, −17.96092179816432974303538685242, −17.14493146806614924050301264837, −16.27436483130287947400928916641, −15.49344743084555407822382774341, −14.83920618520096698411527257381, −14.42894086758327512770466071811, −13.74607685199108315357367774872, −12.73000431262147822064646868938, −12.26823804385208654531530531429, −11.57697632654859112785686108742, −10.51860478761579441265713445804, −9.76557670984418666533615293540, −8.9070778617997622720325147685, −8.77513022976007608541647603993, −7.52063715504093592226799852763, −7.22648748862501839097960555605, −6.213740834349421086753366282296, −5.13578190349665520603699865972, −4.54517540536861818416652490697, −3.56443099951485885157655272351, −2.83682092907659650730671853850, −1.795330632403863055551591963067, −1.50876335679308102971741766578, 0.78623723362241314967052496651, 1.47840459165620017967464653993, 2.72896771557007656231477561657, 3.288383299098109754271950648505, 4.04576439622733579134615272288, 4.884568637890767397180696993164, 5.75154780938946090411789806791, 6.80197375805847856707187690830, 7.556546198816195386609846115367, 8.19000058555893784486414368110, 8.63876818336230617768655997099, 9.867958945092200118343581804, 10.07583346198640901252642533270, 11.10807736220895570264314532944, 11.69730527595022552699319599741, 12.96442128989982754300383465752, 13.26199135561545445969962933261, 14.05467782267310790879968176365, 14.68773666425970263988668058098, 15.16593405269892835913600749224, 16.15362449955928722490103154290, 16.7570144508948527567207970307, 17.448728485608617506979395461036, 18.382308839572817779279403440885, 19.00320424116353762006001339640

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