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Properties

Label 2-4032-48.35-c1-0-17
Degree $2$
Conductor $4032$
Sign $0.0830 - 0.996i$
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  + (2.28 + 2.28i)5-s + 7-s + (3.92 − 3.92i)11-s + (1.38 + 1.38i)13-s + 4.10i·17-s + (−3.60 + 3.60i)19-s + 2.68i·23-s + 5.48i·25-s + (−6.10 + 6.10i)29-s + 5.45i·31-s + (2.28 + 2.28i)35-s + (−4.04 + 4.04i)37-s − 7.75·41-s + (−0.0577 − 0.0577i)43-s − 1.70·47-s + ⋯
L(s)  = 1  + (1.02 + 1.02i)5-s + 0.377·7-s + (1.18 − 1.18i)11-s + (0.385 + 0.385i)13-s + 0.995i·17-s + (−0.827 + 0.827i)19-s + 0.559i·23-s + 1.09i·25-s + (−1.13 + 1.13i)29-s + 0.980i·31-s + (0.387 + 0.387i)35-s + (−0.665 + 0.665i)37-s − 1.21·41-s + (−0.00879 − 0.00879i)43-s − 0.248·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.0830 - 0.996i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.0830 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.435925946\)
\(L(\frac12)\) \(\approx\) \(2.435925946\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 + (-2.28 - 2.28i)T + 5iT^{2} \)
11 \( 1 + (-3.92 + 3.92i)T - 11iT^{2} \)
13 \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \)
17 \( 1 - 4.10iT - 17T^{2} \)
19 \( 1 + (3.60 - 3.60i)T - 19iT^{2} \)
23 \( 1 - 2.68iT - 23T^{2} \)
29 \( 1 + (6.10 - 6.10i)T - 29iT^{2} \)
31 \( 1 - 5.45iT - 31T^{2} \)
37 \( 1 + (4.04 - 4.04i)T - 37iT^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 + (0.0577 + 0.0577i)T + 43iT^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 + (0.517 + 0.517i)T + 53iT^{2} \)
59 \( 1 + (-7.10 + 7.10i)T - 59iT^{2} \)
61 \( 1 + (-2.19 - 2.19i)T + 61iT^{2} \)
67 \( 1 + (-6.98 + 6.98i)T - 67iT^{2} \)
71 \( 1 - 0.356iT - 71T^{2} \)
73 \( 1 + 7.05iT - 73T^{2} \)
79 \( 1 - 0.163iT - 79T^{2} \)
83 \( 1 + (7.33 + 7.33i)T + 83iT^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 - 8.56T + 97T^{2} \)
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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

−8.747823249046151115137367652726, −7.967196314724585332041598767938, −6.84361940849723362261707315016, −6.41972748466341676161490481757, −5.87810777139951768932776652085, −5.02627294688486793223547472170, −3.57280529630606312296195123949, −3.50859966207428969993773810412, −1.97010553990695671346590248289, −1.47349689646537342137249765794, 0.67943480587047176798727022416, 1.78970294403405661382563977637, 2.36653544315862899845956950904, 3.88925287831156527238786160168, 4.56604668474216283729026530846, 5.23145424022125527594295260194, 5.98482969402083187954115269768, 6.78978241912753296264929512371, 7.46693032928072851865851780201, 8.557825717122874421811141034036

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