Properties

Label 2-4032-56.27-c1-0-75
Degree $2$
Conductor $4032$
Sign $-0.707 + 0.707i$
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  + 2.16·5-s − 2.64i·7-s − 4.55·13-s + 0.979i·19-s − 6i·23-s − 0.291·25-s − 5.74i·35-s − 7.00·49-s − 14.4i·59-s − 15.6·61-s − 9.87·65-s − 15.8i·71-s − 5.29i·79-s + 1.40i·83-s + 12.0i·91-s + ⋯
L(s)  = 1  + 0.970·5-s − 0.999i·7-s − 1.26·13-s + 0.224i·19-s − 1.25i·23-s − 0.0583·25-s − 0.970i·35-s − 49-s − 1.87i·59-s − 1.99·61-s − 1.22·65-s − 1.88i·71-s − 0.595i·79-s + 0.153i·83-s + 1.26i·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215027629\)
\(L(\frac12)\) \(\approx\) \(1.215027629\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT \)
good5 \( 1 - 2.16T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 0.979iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 + 15.6T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 - 1.40iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
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

−8.010088833610920260965459342276, −7.47516251784418575054985429983, −6.61683594160161620479691424151, −6.10067829703490184674252945341, −5.03384397825934031104576656439, −4.54975056668956547417748182751, −3.48795091308810697470708076986, −2.48734906661981848553451924962, −1.64712820473225209484076140818, −0.31511251794058783961764581973, 1.50641882092080205563680983455, 2.36353364053478344928124147467, 3.01558116658420293503477746332, 4.27467357553202860653187387594, 5.24930798627245230028300378900, 5.62273890272500780600559326941, 6.40377594234714886705949677099, 7.27275189219451302055615714899, 7.947657376699942392036489997587, 8.951651766852160218405557864697

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