Properties

Label 2-3744-104.77-c1-0-62
Degree $2$
Conductor $3744$
Sign $-0.759 + 0.650i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.66·5-s + 3.57i·7-s − 1.02·11-s + (1.87 − 3.07i)13-s − 5.05·17-s − 1.16·19-s − 8.17·23-s − 2.23·25-s − 4.29i·29-s − 7.98i·31-s + 5.93i·35-s − 9.83·37-s + 1.62i·41-s + 2.35i·43-s − 7.73i·47-s + ⋯
L(s)  = 1  + 0.743·5-s + 1.35i·7-s − 0.309·11-s + (0.520 − 0.853i)13-s − 1.22·17-s − 0.266·19-s − 1.70·23-s − 0.447·25-s − 0.798i·29-s − 1.43i·31-s + 1.00i·35-s − 1.61·37-s + 0.253i·41-s + 0.358i·43-s − 1.12i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3700331559\)
\(L(\frac12)\) \(\approx\) \(0.3700331559\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 3.07i)T \)
good5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 - 3.57iT - 7T^{2} \)
11 \( 1 + 1.02T + 11T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 + 4.29iT - 29T^{2} \)
31 \( 1 + 7.98iT - 31T^{2} \)
37 \( 1 + 9.83T + 37T^{2} \)
41 \( 1 - 1.62iT - 41T^{2} \)
43 \( 1 - 2.35iT - 43T^{2} \)
47 \( 1 + 7.73iT - 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 - 4.41iT - 73T^{2} \)
79 \( 1 - 6.94T + 79T^{2} \)
83 \( 1 + 2.29T + 83T^{2} \)
89 \( 1 + 9.02iT - 89T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−8.280279548606635937630868229264, −7.69750999823789497811765734004, −6.39387655987696423422136714144, −6.00752586986815843959739075153, −5.44261477504617260827658468315, −4.48325383532361876753940367855, −3.45803023276623709973206090105, −2.29695797841786120946815362154, −1.98913873278759726953898107865, −0.096095575243328341010274785497, 1.45893073111826272470711272288, 2.16610763414508762312846085095, 3.52210334031024839258262357697, 4.16827240197500294320290699406, 4.95713826757519131970446449965, 5.93368276828263308046740065215, 6.70173804448100577738530620851, 7.11050344898888606920162602156, 8.123408639553412245879740564778, 8.796658811288974348292446120600

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