L-function 1-3040-3040.1133-r0-0-0

• Label: 1-3040-3040.1133-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.0623 - 0.998i$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0623 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.0623 - 0.998i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.0623 - 0.998i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3887617283 - 0.4138175805i$
$L(1)$	$\approx$	$0.6455101581 - 0.1641955029i$

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

−19.2300731451841349477637435044, −18.66526276262458352863825165722, −17.59232456557549975005991702824, −17.03447651080103768690899877849, −16.31059510197884284465814371346, −15.86564677172936277350292311934, −15.1474329631082714594730008590, −14.46303222872366562624563574762, −13.63905696825716313034849863680, −12.75891436025802654915843077857, −12.29415557364367197896335076674, −11.04426865581638006290305496170, −10.87254441014262894618058772189, −9.9250782196608043493595527749, −9.31709875216364385332514311175, −8.75144116387156489453422780191, −7.7368306001446051731232629022, −6.83083023846682068077063600737, −5.96597637315093534480275677670, −5.496250587390188086891994518724, −4.39409006668894139749099773309, −3.94872649995711636239474323749, −2.82020311601653832979508393201, −2.421093231783698395742989316036, −0.59230716530737870199932566384, 0.29414679665039455041151220226, 1.52423516464448286073111850936, 2.546110331664498310157678202255, 2.92363398073202539082322151607, 4.20644310037598101007914380613, 5.20753116911915996686759400761, 5.73385698729389628833648064772, 6.78460333619847021345216413504, 7.23163724293241345587351483259, 7.771276589287337936252824624202, 8.94757100204563084112854641578, 9.48299504746713556126451319071, 10.41680740429777445324174838265, 11.088660733439761583676737748991, 12.04078186952140637091540475, 12.5517640765955727133056586337, 13.20531183029794042783066192139, 13.6096086095329634582101228385, 14.74731544136244478404446576678, 15.269606461851755533867612791242, 16.274802727956066901508475460699, 16.72956900889852859653645107560, 17.77188562127573889447160659444, 17.98338231076729771309265448987, 18.866674416567196782459687823949

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