Properties

Label 2-3744-1.1-c1-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s + 2·11-s − 13-s − 6·17-s + 2·19-s − 25-s − 2·29-s − 2·31-s − 4·35-s + 10·37-s − 2·41-s − 8·43-s − 2·47-s − 3·49-s + 2·53-s − 4·55-s + 10·59-s + 6·61-s + 2·65-s − 14·67-s − 14·71-s − 6·73-s + 4·77-s + 4·79-s + 6·83-s + 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.277·13-s − 1.45·17-s + 0.458·19-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.676·35-s + 1.64·37-s − 0.312·41-s − 1.21·43-s − 0.291·47-s − 3/7·49-s + 0.274·53-s − 0.539·55-s + 1.30·59-s + 0.768·61-s + 0.248·65-s − 1.71·67-s − 1.66·71-s − 0.702·73-s + 0.455·77-s + 0.450·79-s + 0.658·83-s + 1.30·85-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: \(1\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.129420335341496676740588750751, −7.46151031860632401353885936342, −6.80853739566215081891328748358, −5.93950684715358742645951935079, −4.91766131446373690221625433565, −4.32623638903405619197803847909, −3.60988191855399388392801369311, −2.47201184061771210255460683617, −1.42822565944849691945661197812, 0, 1.42822565944849691945661197812, 2.47201184061771210255460683617, 3.60988191855399388392801369311, 4.32623638903405619197803847909, 4.91766131446373690221625433565, 5.93950684715358742645951935079, 6.80853739566215081891328748358, 7.46151031860632401353885936342, 8.129420335341496676740588750751

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