Properties

Label 2-4032-28.27-c1-0-43
Degree $2$
Conductor $4032$
Sign $0.755 + 0.654i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + 3.46i·5-s + (−2 − 1.73i)7-s − 3.46i·11-s + 3.46i·13-s − 2·19-s − 3.46i·23-s − 6.99·25-s + 6·29-s − 8·31-s + (5.99 − 6.92i)35-s + 2·37-s − 6.92i·41-s − 10.3i·43-s + (1.00 + 6.92i)49-s + 6·53-s + ⋯
L(s)  = 1  + 1.54i·5-s + (−0.755 − 0.654i)7-s − 1.04i·11-s + 0.960i·13-s − 0.458·19-s − 0.722i·23-s − 1.39·25-s + 1.11·29-s − 1.43·31-s + (1.01 − 1.17i)35-s + 0.328·37-s − 1.08i·41-s − 1.58i·43-s + (0.142 + 0.989i)49-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.203222481\)
\(L(\frac12)\) \(\approx\) \(1.203222481\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−8.438706800639572008712255992784, −7.19223352513606076732536581838, −7.04821121628172993232073123278, −6.27874542988212070303393365281, −5.67239664290434402623655635383, −4.29330293750650602807929625854, −3.64549085750344638659901625137, −2.94825323482482435308327660990, −2.07195135518234775470214142646, −0.41472991974883482330920704373, 0.926629528666825442809814410938, 2.00298502874942725079083606883, 3.03635093879846425250562422419, 4.07564610991572297783634717864, 4.87582852841394568726159052194, 5.44434021201073622800100497731, 6.17887122719955139873575357224, 7.09734154702325248906431418607, 8.019826144864740407884777297903, 8.475164310589725539535958914352

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