Properties

Label 2-3744-1.1-c1-0-22
Degree $2$
Conductor $3744$
Sign $1$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·11-s + 13-s + 4·19-s + 4·23-s − 25-s − 8·29-s + 8·31-s + 2·37-s + 6·41-s − 4·43-s + 6·47-s − 7·49-s − 4·53-s + 4·55-s + 6·59-s + 2·61-s + 2·65-s + 4·67-s + 6·71-s − 2·73-s − 16·79-s + 2·83-s + 10·89-s + 8·95-s − 2·97-s + 8·101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s + 0.277·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.549·53-s + 0.539·55-s + 0.781·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.712·71-s − 0.234·73-s − 1.80·79-s + 0.219·83-s + 1.05·89-s + 0.820·95-s − 0.203·97-s + 0.796·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3744} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526748104\)
\(L(\frac12)\) \(\approx\) \(2.526748104\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.609685173226179453665442258483, −7.74652499317951017964613601700, −6.98814297626254924193548047807, −6.20778793559778146432602907404, −5.62349404399732664506023989791, −4.81281047705604358963746290044, −3.84357586762384102764374571608, −2.94793843774524120975754866963, −1.93838116414026850576262121415, −0.979418705281808469159785692374, 0.979418705281808469159785692374, 1.93838116414026850576262121415, 2.94793843774524120975754866963, 3.84357586762384102764374571608, 4.81281047705604358963746290044, 5.62349404399732664506023989791, 6.20778793559778146432602907404, 6.98814297626254924193548047807, 7.74652499317951017964613601700, 8.609685173226179453665442258483

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