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Properties

Label 2-3744-39.23-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.996 - 0.0789i$
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  + 0.517·5-s + (0.866 − 0.5i)13-s + (1.67 + 0.965i)17-s − 0.732·25-s + (−1.67 + 0.965i)29-s + (1.5 − 0.866i)37-s + (0.258 + 0.448i)41-s + (−0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.448 − 0.258i)65-s + i·73-s + (0.866 + 0.499i)85-s + (0.707 + 1.22i)89-s + (1.73 + i)97-s + ⋯
L(s)  = 1  + 0.517·5-s + (0.866 − 0.5i)13-s + (1.67 + 0.965i)17-s − 0.732·25-s + (−1.67 + 0.965i)29-s + (1.5 − 0.866i)37-s + (0.258 + 0.448i)41-s + (−0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.448 − 0.258i)65-s + i·73-s + (0.866 + 0.499i)85-s + (0.707 + 1.22i)89-s + (1.73 + i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.996 - 0.0789i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.996 - 0.0789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530009965\)
\(L(\frac12)\) \(\approx\) \(1.530009965\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 - 0.517T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.517iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.631262262596583114589580069693, −7.958771787002105490510112953919, −7.37277431905176974379546025771, −6.24929668905232549805560323085, −5.78072401225649609387975020133, −5.15419006239725740822202104797, −3.84966105601654730415197075797, −3.40313876373107332395890197335, −2.13395882685487216193525138293, −1.19275030427299492272946072231, 1.12310692478247259142339992122, 2.17859077986770378597526292831, 3.21550868631154813511820473576, 4.02015969861247860335737717910, 4.97737372285398679915660006430, 5.91005595911086490354529832731, 6.16736959086043832302306796245, 7.46183279733293350979282022048, 7.73134109596296483809018325989, 8.785786243361850596281900316598

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