Properties

Degree $2$
Conductor $3744$
Sign $-0.939 - 0.342i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.766 − 1.32i)5-s + (0.173 − 0.300i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)13-s + (1.43 + 0.524i)15-s − 1.87·17-s + (0.0603 + 0.342i)21-s + (−0.673 + 1.16i)25-s + (0.500 + 0.866i)27-s + (−0.5 − 0.866i)31-s − 0.532·35-s + 1.53·37-s + (0.939 + 0.342i)39-s + (−0.939 + 1.62i)43-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.766 − 1.32i)5-s + (0.173 − 0.300i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)13-s + (1.43 + 0.524i)15-s − 1.87·17-s + (0.0603 + 0.342i)21-s + (−0.673 + 1.16i)25-s + (0.500 + 0.866i)27-s + (−0.5 − 0.866i)31-s − 0.532·35-s + 1.53·37-s + (0.939 + 0.342i)39-s + (−0.939 + 1.62i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.939 - 0.342i$
Motivic weight: \(0\)
Character: $\chi_{3744} (3535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09025218257\)
\(L(\frac12)\) \(\approx\) \(0.09025218257\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.87T + T^{2} \)
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 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
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

−8.201751461680250141829090181963, −7.76556600650504329296894993653, −6.70072109287127429419603478457, −5.92213932929309779163457293428, −5.03293883619685911360967087803, −4.47549417812639687870051519295, −4.07772831943094740135274776692, −2.80871150141933921927577870020, −1.22952677383235132616985765277, −0.06019349182816634045108503560, 1.92695788602108619953694636416, 2.55678852059728142736891584574, 3.75896972628953209098223509902, 4.58032883635711901996472717360, 5.41664100408565716640796545973, 6.54456496789950237266869956467, 6.78984792652323577775899893505, 7.34027301904693615550156892369, 8.234166575522716072666202937586, 8.936638812272539007623976326103

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