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Properties

Label 2-3648-228.227-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9309508772\)
\(L(\frac12)\) \(\approx\) \(0.9309508772\)
\(L(1)\) 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

−8.833474514019028165631854262030, −7.78085443223989364892979048410, −7.13465297620706896822012355749, −6.54496599004501843899773637369, −5.51211887913210021354246859911, −5.21904491271008390608688473866, −4.17810961685127582492727153722, −3.36741983337435788778300682590, −2.07505873969819187743029770262, −0.899567923188415075070838756471, 0.899567923188415075070838756471, 2.07505873969819187743029770262, 3.36741983337435788778300682590, 4.17810961685127582492727153722, 5.21904491271008390608688473866, 5.51211887913210021354246859911, 6.54496599004501843899773637369, 7.13465297620706896822012355749, 7.78085443223989364892979048410, 8.833474514019028165631854262030

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