Properties

Label 2-3744-13.11-c0-0-2
Degree $2$
Conductor $3744$
Sign $-0.679 + 0.733i$
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  + (−0.366 − 0.366i)5-s + (−0.866 + 0.5i)13-s + (−1.5 − 0.866i)17-s − 0.732i·25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.133i)37-s + (0.5 − 1.86i)41-s + (0.866 − 0.5i)49-s − 1.73·53-s + (−0.5 + 0.866i)61-s + (0.5 + 0.133i)65-s + (−0.366 + 0.366i)73-s + (0.232 + 0.866i)85-s + (−1.36 − 0.366i)89-s + (1.36 − 0.366i)97-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)5-s + (−0.866 + 0.5i)13-s + (−1.5 − 0.866i)17-s − 0.732i·25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.133i)37-s + (0.5 − 1.86i)41-s + (0.866 − 0.5i)49-s − 1.73·53-s + (−0.5 + 0.866i)61-s + (0.5 + 0.133i)65-s + (−0.366 + 0.366i)73-s + (0.232 + 0.866i)85-s + (−1.36 − 0.366i)89-s + (1.36 − 0.366i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5451760554\)
\(L(\frac12)\) \(\approx\) \(0.5451760554\)
\(L(1)\) 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 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.552967682025295553220710667839, −7.56539721130351316671032109767, −7.07667809457908296256118972540, −6.26163185510634858362528600701, −5.30634038482444675740993940029, −4.52416585693641884165444578135, −4.00577669416578702898778452006, −2.71500682773321181389534093393, −1.97193332566613469593295784675, −0.29244452085144354466148667710, 1.60512409459421555681250842975, 2.67632431777107736698824264420, 3.48863486579827925638275280541, 4.43275356600897654752548610429, 5.11283148023672263099272972332, 6.11671430373085885341729484410, 6.77070027555303649423704147599, 7.53301772055677269152031115078, 8.118367822036802416857697135949, 9.000555995460260943396256960220

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