L-function 2-2100-84.83-c0-0-3

• Label: 2-2100-84.83-c0-0-3
• Degree: $2$
• Conductor: $2100$
• Sign: $1$
• Analytic cond.: $1.04803$
• Root an. cond.: $1.02373$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 2^-s + 3^-s + 4^-s + 6^-s − 7^-s + 8^-s + 9^-s + 12^-s − 14^-s + 16^-s + 18^-s − 21^-s − 2·23^-s + 24^-s + 27^-s − 28^-s + 32^-s + 36^-s − 2·41^-s − 42^-s − 2·46^-s + 48^-s + 49^-s + 54^-s − 56^-s − 63^-s + 64^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	\(2\)
Conductor:	\(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign:	$1$
Analytic conductor:	\(1.04803\)
Root analytic conductor:	\(1.02373\)
Motivic weight:	\(0\)
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{2100} (251, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	\(0\)
Selberg data:	\((2,\ 2100,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(2.811044248\)
\(L(1)\)		not available

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−9.381101828112642004371091138532, −8.355227375452525759881372702682, −7.69375659327292345746089966981, −6.80355306498114660609884852731, −6.24310554529546923996267413298, −5.22213240619519963076035493607, −4.12507082419648520257472439400, −3.57198073261780504216600819425, −2.70745426431371197704736828299, −1.78145721596023673987688067974, 1.78145721596023673987688067974, 2.70745426431371197704736828299, 3.57198073261780504216600819425, 4.12507082419648520257472439400, 5.22213240619519963076035493607, 6.24310554529546923996267413298, 6.80355306498114660609884852731, 7.69375659327292345746089966981, 8.355227375452525759881372702682, 9.381101828112642004371091138532

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