Properties

Label 2-3744-936.571-c0-0-2
Degree $2$
Conductor $3744$
Sign $0.173 - 0.984i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)3-s + (0.173 − 0.300i)5-s + (0.939 + 1.62i)7-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)13-s + (0.266 + 0.223i)15-s + 1.53·17-s + (−1.76 + 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (0.5 − 0.866i)31-s + 0.652·35-s − 0.347·37-s + (0.766 + 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (0.173 − 0.300i)5-s + (0.939 + 1.62i)7-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)13-s + (0.266 + 0.223i)15-s + 1.53·17-s + (−1.76 + 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (0.5 − 0.866i)31-s + 0.652·35-s − 0.347·37-s + (0.766 + 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.460909071\)
\(L(\frac12)\) \(\approx\) \(1.460909071\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.347T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.87T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.869782350827012680799101128058, −8.263363872179103464570389352578, −7.73083928030311701602512651113, −6.26614317324476149641010179359, −5.48722068134373416429935480604, −5.37679043719308304828539637669, −4.44570526440947321995955933817, −3.33257256759645255391418777525, −2.65603964865468624861587147953, −1.36867557474791387746263336947, 1.03930106763587803135405709687, 1.65722199959778919819522399452, 2.93563920650582969014263299834, 3.91190634924372386119770913778, 4.74984429405780828159917966105, 5.62975017687288098504407926078, 6.55850592848577855801862551540, 7.05214633578318266202999216527, 7.74161543219726980898370135041, 8.223782498217295097326990992817

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