L-function 2-30e2-100.19-c0-0-1

• Label: 2-30e2-100.19-c0-0-1
• Degree: $2$
• Conductor: $900$
• Sign: $0.968 + 0.248i$
• Analytic cond.: $0.449158$
• Root an. cond.: $0.670192$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.951 − 0.309i)2^-s + (0.809 − 0.587i)4^-s + (−0.587 + 0.809i)5^-s + (0.587 − 0.809i)8^-s + (−0.309 + 0.951i)10^-s + (1.11 + 0.363i)13^-s + (0.309 − 0.951i)16^-s + (0.363 − 0.5i)17^-s + 0.999i·20^-s + (−0.309 − 0.951i)25^-s + 1.17·26^-s + (−1.53 + 1.11i)29^-s − i·32^-s + (0.190 − 0.587i)34^-s + (−1.80 − 0.587i)37^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$900$    =    $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.968 + 0.248i$
Analytic conductor:	$0.449158$
Root analytic conductor:	$0.670192$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (19, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :0),\ 0.968 + 0.248i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.644352888$
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$1 + (-0.951 + 0.309i)T$
	3	$1$
	5	$1 + (0.587 - 0.809i)T$
good	7	$1 + T^{2}$
	11	$1 + (0.809 - 0.587i)T^{2}$
	13	$1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2}$
	17	$1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}$
	19	$1 + (-0.309 - 0.951i)T^{2}$
	23	$1 + (-0.809 + 0.587i)T^{2}$
	29	$1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2}$
	31	$1 + (-0.309 - 0.951i)T^{2}$
	37	$1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2}$

Imaginary part of the first few zeros on the critical line

−10.59986900329159309509521787607, −9.699461215020015615865776926570, −8.550558271742737435183097818157, −7.42482082393729436219161993207, −6.79123530658515058052965022890, −5.90737279993285978386263072697, −4.89737206448011356629001860126, −3.72204483274331558858568224127, −3.21130789358404287854821928286, −1.76848825607911662501926024035, 1.71336975960622645738288534676, 3.41174887955700708217934102415, 4.00011315318874437935089740500, 5.12256344707492867425207956684, 5.81573700153551513224530902272, 6.81811971233623963742477683231, 7.895648210737973335928155014108, 8.333636266970893915357082929706, 9.375643454328889216790574393391, 10.66771589065815058168348317787

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