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Properties

Label 2-3648-228.227-c0-0-1
Degree $2$
Conductor $3648$
Sign $1$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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  + 3-s + 9-s − 19-s + 25-s + 27-s + 2·31-s + 49-s − 57-s + 2·61-s − 2·67-s − 2·73-s + 75-s − 2·79-s + 81-s + 2·93-s − 2·103-s + ⋯
L(s)  = 1  + 3-s + 9-s − 19-s + 25-s + 27-s + 2·31-s + 49-s − 57-s + 2·61-s − 2·67-s − 2·73-s + 75-s − 2·79-s + 81-s + 2·93-s − 2·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3648} (3647, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.909344704\)
\(L(\frac12)\) \(\approx\) \(1.909344704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
19 \( 1 + T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.591301961201449829851855204192, −8.216306321911532363916655727111, −7.24710445444653721722607224116, −6.69901101561129395766347266126, −5.79214968530768370379278784896, −4.63729919732659994992185111138, −4.15183464111936224443301202461, −3.05100633811338489925939350046, −2.42991901601560194543710392249, −1.26009431587852064166583954267, 1.26009431587852064166583954267, 2.42991901601560194543710392249, 3.05100633811338489925939350046, 4.15183464111936224443301202461, 4.63729919732659994992185111138, 5.79214968530768370379278784896, 6.69901101561129395766347266126, 7.24710445444653721722607224116, 8.216306321911532363916655727111, 8.591301961201449829851855204192

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