Properties

Label 2-4032-28.27-c1-0-10
Degree $2$
Conductor $4032$
Sign $-0.0839 - 0.996i$
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  + 1.72i·5-s + (−2.63 + 0.222i)7-s − 6.17i·11-s + 2.82i·13-s − 1.72i·17-s − 5.90·19-s − 1.54i·23-s + 2.01·25-s + 8.28·29-s + 4.62·31-s + (−0.384 − 4.55i)35-s − 2.24·37-s + 3.92i·41-s + 10.1i·43-s + 4.88·47-s + ⋯
L(s)  = 1  + 0.772i·5-s + (−0.996 + 0.0839i)7-s − 1.86i·11-s + 0.784i·13-s − 0.419i·17-s − 1.35·19-s − 0.322i·23-s + 0.402·25-s + 1.53·29-s + 0.831·31-s + (−0.0649 − 0.770i)35-s − 0.368·37-s + 0.613i·41-s + 1.54i·43-s + 0.713·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.0839 - 0.996i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.0839 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055039907\)
\(L(\frac12)\) \(\approx\) \(1.055039907\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.222i)T \)
good5 \( 1 - 1.72iT - 5T^{2} \)
11 \( 1 + 6.17iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 1.72iT - 17T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
23 \( 1 + 1.54iT - 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 - 4.62T + 31T^{2} \)
37 \( 1 + 2.24T + 37T^{2} \)
41 \( 1 - 3.92iT - 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 + 9.17T + 53T^{2} \)
59 \( 1 + 9.65T + 59T^{2} \)
61 \( 1 - 14.2iT - 61T^{2} \)
67 \( 1 - 6.64iT - 67T^{2} \)
71 \( 1 + 5.00iT - 71T^{2} \)
73 \( 1 - 2.19iT - 73T^{2} \)
79 \( 1 - 9.55iT - 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 9.16iT - 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−8.672182852198867881270954973931, −8.014309263106639267037091429382, −6.91330314048820272029789656888, −6.29604120645192832296754560253, −6.13878221775430480406672348379, −4.81916467937666900711190603540, −3.94982634658155775597114018725, −2.96546350736797477737306584240, −2.66875701203548890734027180920, −0.976243964121298971844244382193, 0.35496980857737903510868717774, 1.70403341528665529493179221865, 2.64572077959719076071134752861, 3.70364935842498823530593964052, 4.57224455418605531503752946230, 5.06076137656160164732815124015, 6.19583960768583839448899204932, 6.70025363658418664458663847876, 7.52684440837012629655364149668, 8.295209146192539802659283341577

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