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Properties

Label 2-3744-13.7-c0-0-2
Degree $2$
Conductor $3744$
Sign $-0.233 + 0.972i$
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 + (−0.866 + 0.5i)17-s − 0.732i·25-s + (0.866 − 1.5i)29-s + (−0.5 − 1.86i)37-s + (1.86 − 0.5i)41-s + (−0.866 − 0.5i)49-s − 53-s + (−0.5 − 0.866i)61-s + (0.133 + 0.5i)65-s + (−1.36 + 1.36i)73-s + (0.5 + 0.133i)85-s + (0.366 + 1.36i)89-s + (0.366 − 1.36i)97-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)5-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s − 0.732i·25-s + (0.866 − 1.5i)29-s + (−0.5 − 1.86i)37-s + (1.86 − 0.5i)41-s + (−0.866 − 0.5i)49-s − 53-s + (−0.5 − 0.866i)61-s + (0.133 + 0.5i)65-s + (−1.36 + 1.36i)73-s + (0.5 + 0.133i)85-s + (0.366 + 1.36i)89-s + (0.366 − 1.36i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.233 + 0.972i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.233 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8046578055\)
\(L(\frac12)\) \(\approx\) \(0.8046578055\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T + T^{2} \)
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 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.360347785494956549812642523093, −7.85798750052366675773861283517, −7.07908145962316779861360795346, −6.22717817787434151546152209828, −5.49939248862013686052314547409, −4.51160195964428402636625898224, −4.07563316901414771494434895552, −2.83876289848417107042140460102, −2.03066226051271144559346248575, −0.45516746461427546862159337261, 1.45984977015427181914609211515, 2.67290416355580589303592227347, 3.31669762662698611286325057383, 4.54830090088859657648416169167, 4.87748942703355401394298783369, 6.06791579751998446191422484863, 6.81390104417994494345368841526, 7.34108477683715811918947445134, 8.100147193084199351478464979255, 9.003933966836313844423483062417

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