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Properties

Label 2-3744-39.23-c0-0-3
Degree $2$
Conductor $3744$
Sign $0.406 + 0.913i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.406 + 0.913i$
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: \((2,\ 3744,\ (\ :0),\ 0.406 + 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008762622\)
\(L(\frac12)\) \(\approx\) \(1.008762622\)
\(L(1)\) 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

−8.440518081364918706876866614761, −7.958882375737210614775901215856, −7.04988503474475635986668323170, −6.40283784292561223354281977332, −5.61892115700758155073084586800, −4.58383423113128369780035579907, −4.06432717504393786301322810689, −3.01548926698262717699930143396, −2.14664350888057314296599902550, −0.62152294789323721786420235282, 1.30337756213462768639541546582, 2.41670143497501166319090316318, 3.48090386482220058863492382563, 4.28073289943006633439795048330, 4.82894376363242212021326605164, 6.18755711435086319100172054549, 6.41877159784162976571535176022, 7.36599968091897486012933467190, 8.314067877467431242112980023227, 8.594508373316987413779805493102

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