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Properties

Label 2-3744-52.3-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.00641 + 0.999i$
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  + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)23-s − 25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s − 2i·47-s + (0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + 13-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)23-s − 25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s − 2i·47-s + (0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.00641 + 0.999i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.00641 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8710683958\)
\(L(\frac12)\) \(\approx\) \(0.8710683958\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.555810608817816495948121927881, −7.63849531876148730147850068018, −7.11237229156075084724905511128, −6.29738186834124370324267093478, −5.59905036174912904897011108129, −4.65252383435788242339383980389, −3.84873173974687364641393048571, −3.00510465094881321589551489382, −2.07148267056603415708706039810, −0.50456235315438916296005531625, 1.39083069657033425645385176931, 2.60679838786445375692486019532, 3.42801944631849603077841054838, 4.09759718525409104929324204780, 5.41854634854879107883287409494, 5.85099254782685581030434711768, 6.50343341194240928505157241558, 7.47891124015543794052948237128, 8.225464341433303048631784673523, 8.821940590758139830308055955951

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