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Properties

Label 2-3744-468.367-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.985 + 0.167i$
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)3-s + (0.5 + 0.866i)5-s + (0.499 + 0.866i)9-s − i·11-s + (0.5 − 0.866i)13-s − 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999i·27-s + 29-s + (−0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.499i)39-s + (1.73 − i)43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (0.499 + 0.866i)9-s − i·11-s + (0.5 − 0.866i)13-s − 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999i·27-s + 29-s + (−0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (−0.866 + 0.499i)39-s + (1.73 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.985 + 0.167i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.985 + 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096888986\)
\(L(\frac12)\) \(\approx\) \(1.096888986\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + iT - T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.477996407941092657483313976924, −7.895196738801804191995937436321, −7.00527450115587820019037873007, −6.39697870375717573355845413510, −5.75614782996802831822082922882, −5.29556674689736393383725830041, −3.98815288554879997475912574589, −3.12353890809117756110062021008, −2.13136525716493570700301966805, −0.957708135723932222008143593093, 1.00180706140803110803369212066, 2.04435058896969122450530847220, 3.40041043207895757785375674241, 4.52791959318581277282530152215, 4.83126324758970553336794264231, 5.55042011351218709113975657122, 6.53635916722032178099091228685, 6.96549363027541567721350644048, 8.000337934904620036598082634964, 9.058482130748630512174501378480

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