L-function 1-3040-3040.1203-r0-0-0

• Label: 1-3040-3040.1203-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.988 + 0.153i$
• 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

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.988 + 0.153i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1203, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.988 + 0.153i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.01888220686 + 0.2444015111i$
$L(1)$	$\approx$	$0.6513305494 + 0.07169197262i$

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

−18.772708535453452600088203515188, −18.07782974396076690158174628950, −17.14515510908293582671763933306, −16.79654527888701734312891143507, −15.75211997697940772134390460833, −15.60668762424881818870671362247, −14.58494710985790564283993367248, −13.57722669945715238800457361903, −13.03129104490883763169050518722, −12.12790550804418400430609759753, −11.800798854923408041626193323163, −11.03604302546259134453194706116, −10.01593323351968510521873205891, −9.78353386225031362989239631482, −8.57286936969309162026083706537, −7.94535690272254460043010721713, −7.027132819827760201709206611565, −6.06346331862553461611762082314, −5.73962607228276459263109969873, −5.11312838662821925328978407432, −3.9147235492897632890153320695, −3.20541485419797148745680964143, −2.20019587357214708511693769389, −1.08705619657773726428417337094, −0.0990857631164895514789298076, 1.231092793657381316548088968719, 1.88472152483390022122408332979, 3.32664118042003873674814889515, 4.10389897251428143919609195600, 4.67209517845253127791773768436, 5.57468389184959496063295506034, 6.43208951886229857014276566708, 7.047742218943437259501870712672, 7.51964225754101785302198300833, 8.76005058181528320571452517662, 9.698024970761220618940061172789, 10.16903274227337068477104521822, 10.74376167107705500507054480731, 11.765126522473862360979319156731, 12.21698308996364273949184942461, 12.84819844817923812524517721003, 13.78901880492315270891878237063, 14.41156972685883527634375669438, 15.292681239526343084210599562307, 16.169578100207013294017184035180, 16.60704926368406402888108820705, 17.16902864330709440904135797408, 17.868077756778489946535285211, 18.611206902750373470585790457119, 19.27839643172202949302633870183

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