L-function 2-78e2-1.1-c1-0-53

• Label: 2-78e2-1.1-c1-0-53
• Degree: $2$
• Conductor: $6084$
• Sign: $-1$
• Analytic cond.: $48.5809$
• Root an. cond.: $6.97000$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: yes
• Analytic rank: $1$

Dirichlet series

L(s)  = 1

+ 1.73·7^-s − 3.46·19^-s − 5·25^-s + 1.73·31^-s − 6.92·37^-s − 5·43^-s − 4·49^-s + 61^-s − 15.5·67^-s + 15.5·73^-s − 17·79^-s + 5.19·97^-s + 7·103^-s − 12.1·109^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$6084$    =    $2^{2} \cdot 3^{2} \cdot 13^{2}$
Sign:	$-1$
Analytic conductor:	$48.5809$
Root analytic conductor:	$6.97000$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	Trivial
Primitive:	yes
Self-dual:	yes
Analytic rank:	$1$
Selberg data:	$(2,\ 6084,\ (\ :1/2),\ -1)$

Particular Values

$L(1)$	$=$	$0$
$L(\frac{3}{2})$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	13	$1$
good	5	$1 + 5T^{2}$
	7	$1 - 1.73T + 7T^{2}$
	11	$1 + 11T^{2}$
	17	$1 + 17T^{2}$
	19	$1 + 3.46T + 19T^{2}$
	23	$1 + 23T^{2}$
	29	$1 + 29T^{2}$
	31	$1 - 1.73T + 31T^{2}$
	37	$1 + 6.92T + 37T^{2}$

Imaginary part of the first few zeros on the critical line

−7.81766255114161347993477749730, −7.01882926698614787657487486010, −6.31234789372820725681681334745, −5.54638297763503232778644905724, −4.80321683734389458223072062275, −4.11056584032868454838640183450, −3.25188958921145096612546032330, −2.20513435705524486513020177223, −1.44541016615256329290115722772, 0, 1.44541016615256329290115722772, 2.20513435705524486513020177223, 3.25188958921145096612546032330, 4.11056584032868454838640183450, 4.80321683734389458223072062275, 5.54638297763503232778644905724, 6.31234789372820725681681334745, 7.01882926698614787657487486010, 7.81766255114161347993477749730

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