L-function 2-2550-5.4-c1-0-31

• Label: 2-2550-5.4-c1-0-31
• Degree: $2$
• Conductor: $2550$
• Sign: $0.894 + 0.447i$
• Analytic cond.: $20.3618$
• Root an. cond.: $4.51241$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·2^-s − i·3^-s − 4^-s + 6^-s − i·8^-s − 9^-s + 4·11^-s + i·12^-s − 2i·13^-s + 16^-s + i·17^-s − i·18^-s + 4·19^-s + 4i·22^-s − 4i·23^-s − 24^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$2550$    =    $2 \cdot 3 \cdot 5^{2} \cdot 17$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$20.3618$
Root analytic conductor:	$4.51241$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2550} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2550,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.682861287$
$L(\frac{3}{2})$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$1 - iT$
	3	$1 + iT$
	5	$1$
	17	$1 - iT$
good	7	$1 - 7T^{2}$
	11	$1 - 4T + 11T^{2}$
	13	$1 + 2iT - 13T^{2}$
	19	$1 - 4T + 19T^{2}$
	23	$1 + 4iT - 23T^{2}$
	29	$1 + 2T + 29T^{2}$
	31	$1 + 4T + 31T^{2}$
	37	$1 + 6iT - 37T^{2}$
	41	$1 + 10T + 41T^{2}$

Imaginary part of the first few zeros on the critical line

−8.674976872205014579610399513176, −8.035609656333998927155692578483, −7.17973435817003348097957125423, −6.67711219316901086933526183224, −5.84623157877719780983481316184, −5.15512757928069106842408712785, −4.05414912395753202368217812084, −3.23934968937428611353574434117, −1.86874871860044182921554026911, −0.66659293232709081740859685894, 1.11452878732348489724099533534, 2.20686752998281625449359413655, 3.49275402817289348986607960888, 3.86206364322777521341628953009, 4.93038821072975749988228583327, 5.59984040164011358027007334065, 6.68157484600785741437958656523, 7.42637657357619837608742266578, 8.561070558430045159539506325686, 9.102686175362718376924115419268

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