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Properties

Label 2-4032-28.27-c1-0-48
Degree $2$
Conductor $4032$
Sign $1$
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  + 2.44i·5-s + 2.64·7-s − 6.48i·11-s + 7.34i·17-s − 5.29·19-s − 6.48i·23-s − 0.999·25-s + 10.5·31-s + 6.48i·35-s + 8·37-s − 12.2i·41-s + 7.00·49-s + 15.8·55-s + 6.48i·71-s − 17.1i·77-s + ⋯
L(s)  = 1  + 1.09i·5-s + 0.999·7-s − 1.95i·11-s + 1.78i·17-s − 1.21·19-s − 1.35i·23-s − 0.199·25-s + 1.90·31-s + 1.09i·35-s + 1.31·37-s − 1.91i·41-s + 49-s + 2.14·55-s + 0.769i·71-s − 1.95i·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $1$
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: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.145542051\)
\(L(\frac12)\) \(\approx\) \(2.145542051\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T \)
good5 \( 1 - 2.44iT - 5T^{2} \)
11 \( 1 + 6.48iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7.34iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 6.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 6.48iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.44iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.426241167435220494601672763915, −7.949385623067809453925500698663, −6.81463478431653505050039425158, −6.19827364816612283075447825507, −5.76126058235073063800590026815, −4.50819023170234699227523999283, −3.85394391229596440689945421837, −2.89055737892493040800202725800, −2.12522913673909675416089481199, −0.77912361408718446470305494861, 0.960822308585700928598324417373, 1.85889090997514247765741724530, 2.74675619590804816107286643587, 4.31741667623746288891483386290, 4.69451597921007789672670211398, 5.08669013305805620823043701770, 6.23671379916572919832305282893, 7.15991334030946183114021035053, 7.76484550526882423172114011244, 8.350592759652075728400509246443

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