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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 $0$

Origins

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Normalization:  

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 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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

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