Properties

Label 2-3648-1.1-c1-0-10
Degree $2$
Conductor $3648$
Sign $1$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.56·5-s − 2.56·7-s + 9-s + 1.43·11-s + 5.12·13-s − 2.56·15-s − 5.68·17-s + 19-s − 2.56·21-s − 0.876·23-s + 1.56·25-s + 27-s − 8.24·29-s + 2·31-s + 1.43·33-s + 6.56·35-s + 8·37-s + 5.12·39-s + 3.12·41-s + 2.56·43-s − 2.56·45-s − 5.68·47-s − 0.438·49-s − 5.68·51-s + 12.2·53-s − 3.68·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.14·5-s − 0.968·7-s + 0.333·9-s + 0.433·11-s + 1.42·13-s − 0.661·15-s − 1.37·17-s + 0.229·19-s − 0.558·21-s − 0.182·23-s + 0.312·25-s + 0.192·27-s − 1.53·29-s + 0.359·31-s + 0.250·33-s + 1.10·35-s + 1.31·37-s + 0.820·39-s + 0.487·41-s + 0.390·43-s − 0.381·45-s − 0.829·47-s − 0.0626·49-s − 0.796·51-s + 1.68·53-s − 0.496·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(29.1294\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.515463534\)
\(L(\frac12)\) \(\approx\) \(1.515463534\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
23 \( 1 + 0.876T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 + 5.68T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.494755699730243420436359106258, −7.925979162365744566404471856313, −7.05190504934706925160018075473, −6.49064206178081629712271145289, −5.66785043080806094337111883537, −4.28500090105027440558829668658, −3.89010178800841060133942178064, −3.22132178332037100945954163269, −2.11993624268248353094664287619, −0.68856560986871224591735347460, 0.68856560986871224591735347460, 2.11993624268248353094664287619, 3.22132178332037100945954163269, 3.89010178800841060133942178064, 4.28500090105027440558829668658, 5.66785043080806094337111883537, 6.49064206178081629712271145289, 7.05190504934706925160018075473, 7.925979162365744566404471856313, 8.494755699730243420436359106258

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