L-function 1-3040-3040.1603-r0-0-0

• Label: 1-3040-3040.1603-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $0.339 - 0.940i$
• 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

+ (0.258 + 0.965i)3^-s + 7^-s + (−0.866 + 0.5i)9^-s + (−0.707 + 0.707i)11^-s + (−0.965 − 0.258i)13^-s + (−0.866 − 0.5i)17^-s + (0.258 + 0.965i)21^-s + (−0.5 − 0.866i)23^-s + (−0.707 − 0.707i)27^-s + (−0.258 + 0.965i)29^-s − 31^-s + (−0.866 − 0.5i)33^-s + (0.707 + 0.707i)37^-s − i·39^-s + (−0.866 − 0.5i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$0.339 - 0.940i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1603, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ 0.339 - 0.940i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5206312864 - 0.3656551643i$
$L(1)$	$\approx$	$0.8972323883 + 0.2379339021i$

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	$1 + (0.258 + 0.965i)T$
	7	$1 + T$
	11	$1 + (-0.707 + 0.707i)T$
	13	$1 + (-0.965 - 0.258i)T$
	17	$1 + (-0.866 - 0.5i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (-0.258 + 0.965i)T$
	31	$1 - T$
	37	$1 + (0.707 + 0.707i)T$

Imaginary part of the first few zeros on the critical line

−19.16241809913690505143879986265, −18.402480352131280912762862744972, −17.881671730651833156812224280276, −17.26297435327815559230908731352, −16.58032696230903654728601786598, −15.50486192164053998397517065044, −14.853176387867043233527137230190, −14.22564342059575647270358642751, −13.504973559021119744669308133536, −12.98199132789938675464489480913, −12.107929335585584697433247171123, −11.42211830678607323338914591558, −10.9420773414180795470585845822, −9.869797136301069146145578769, −8.95683834717213647935892005112, −8.30646619757575633207149356173, −7.633583960184287843032565241786, −7.162268694884283062008758540408, −6.03668350055943569997160785186, −5.52163750365518344821702755153, −4.54978046302637657476002817224, −3.63781793718226064977236279095, −2.48124484396294867083388948631, −2.07657834972848243069741233527, −1.0746813537308968279411867846, 0.18137367803221059009588473625, 1.9270897306531035675326465035, 2.41595409652871788546476312517, 3.404480109324258684999602424415, 4.44327097960613524523791662575, 4.90846360943208005776683281135, 5.40746118770996434434946762497, 6.659845801220396281745275831817, 7.58574214251201929375111819299, 8.13651855282680709953677582958, 8.97230786690852187162210618960, 9.65307621932029251086638002818, 10.42790328989838027720105335769, 10.94083972400298876315405860957, 11.70670759002855669978973005818, 12.53975440083223720215611116250, 13.34762381730095770680165418102, 14.37665735850164936449399693994, 14.63629907758277934974236630721, 15.37005081070105645700279262139, 16.01049801487803763710984212846, 16.78142495757680166704734205776, 17.53324405414937795457311603819, 18.04706080828723796692529853077, 18.859780672912939855511895698105

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