Properties

Label 2-4032-24.11-c1-0-3
Degree $2$
Conductor $4032$
Sign $-0.985 + 0.169i$
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  + i·7-s + 4.24i·11-s + 6i·13-s − 4.24·23-s − 5·25-s − 4.24·29-s + 4i·31-s − 6i·37-s − 8.48i·41-s − 6·43-s − 8.48·47-s − 49-s + 12.7·53-s − 8.48i·59-s − 6i·61-s + ⋯
L(s)  = 1  + 0.377i·7-s + 1.27i·11-s + 1.66i·13-s − 0.884·23-s − 25-s − 0.787·29-s + 0.718i·31-s − 0.986i·37-s − 1.32i·41-s − 0.914·43-s − 1.23·47-s − 0.142·49-s + 1.74·53-s − 1.10i·59-s − 0.768i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5046993528\)
\(L(\frac12)\) \(\approx\) \(0.5046993528\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + 8.48iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + 16.9iT - 83T^{2} \)
89 \( 1 + 8.48iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.882573728797241316488891784238, −8.121038423024082749760834662193, −7.20020020880583669837973944067, −6.80142377286183488941458998056, −5.87472715297778390839506675012, −5.06279258406415346869821531979, −4.24225296431010947518190985975, −3.62116990776835524145782518484, −2.10110872134390969165841895213, −1.86315077615667525680878371579, 0.14228804874971602617736121744, 1.23445644086025279572211032659, 2.59418935690332892199898002178, 3.40449678285312499520422442216, 4.08020597178771150561004771456, 5.25287280722824407309816137669, 5.79252410295276734172617003258, 6.46384623162255478689977856472, 7.50768685595505047311050377660, 8.140630737945495921050259538070

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