# Properties

 Label 1-264-264.197-r0-0-0 Degree $1$ Conductor $264$ Sign $1$ Analytic cond. $1.22601$ Root an. cond. $1.22601$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + 5-s − 7-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s + 31-s − 35-s − 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 65-s − 67-s − 71-s − 73-s − 79-s − 83-s + 85-s − 89-s − 91-s + ⋯
 L(s)  = 1 + 5-s − 7-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s + 31-s − 35-s − 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 65-s − 67-s − 71-s − 73-s − 79-s − 83-s + 85-s − 89-s − 91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$264$$    =    $$2^{3} \cdot 3 \cdot 11$$ Sign: $1$ Analytic conductor: $$1.22601$$ Root analytic conductor: $$1.22601$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{264} (197, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 264,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.389934014$$ $$L(\frac12)$$ $$\approx$$ $$1.389934014$$ $$L(1)$$ $$\approx$$ $$1.198289577$$ $$L(1)$$ $$\approx$$ $$1.198289577$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
11 $$1$$
good5 $$1 + T$$
7 $$1 - T$$
13 $$1 + T$$
17 $$1 + T$$
19 $$1 + T$$
23 $$1 - T$$
29 $$1 - T$$
31 $$1 + T$$
37 $$1 - T$$
41 $$1 + T$$
43 $$1 + T$$
47 $$1 - T$$
53 $$1 + T$$
59 $$1 + T$$
61 $$1 + T$$
67 $$1 - T$$
71 $$1 - T$$
73 $$1 - T$$
79 $$1 - T$$
83 $$1 - T$$
89 $$1 - T$$
97 $$1 + T$$
show more
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−25.92648558390805683872667578406, −25.06097349074106557162678051770, −24.122326946002510738419923827736, −22.85755538151709044576609196434, −22.33416689664896284937852727461, −21.18444000985899196111903588413, −20.52679491550270771222564740174, −19.31265843707793477589885707398, −18.43477375556445174306726843021, −17.5873480571489122996940126966, −16.44734217366029771703912066122, −15.85185293812834005886056124938, −14.41533763561748971102289945787, −13.606070877276293607517019559947, −12.82737654800779313730077672825, −11.7035032142437087489452117891, −10.325478105545099956581937529460, −9.71391550926765750108355507425, −8.704704090413726301686463520259, −7.317228757111682871918791963493, −6.13212483464472689841152955043, −5.52676613743764408283123516050, −3.82663000840090336194294558388, −2.753358887465646887815564203757, −1.28386484190770504592501028245, 1.28386484190770504592501028245, 2.753358887465646887815564203757, 3.82663000840090336194294558388, 5.52676613743764408283123516050, 6.13212483464472689841152955043, 7.317228757111682871918791963493, 8.704704090413726301686463520259, 9.71391550926765750108355507425, 10.325478105545099956581937529460, 11.7035032142437087489452117891, 12.82737654800779313730077672825, 13.606070877276293607517019559947, 14.41533763561748971102289945787, 15.85185293812834005886056124938, 16.44734217366029771703912066122, 17.5873480571489122996940126966, 18.43477375556445174306726843021, 19.31265843707793477589885707398, 20.52679491550270771222564740174, 21.18444000985899196111903588413, 22.33416689664896284937852727461, 22.85755538151709044576609196434, 24.122326946002510738419923827736, 25.06097349074106557162678051770, 25.92648558390805683872667578406