Properties

Label 2-4032-48.35-c1-0-42
Degree $2$
Conductor $4032$
Sign $-0.877 + 0.479i$
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  + (1.18 + 1.18i)5-s + 7-s + (−1.81 + 1.81i)11-s + (−3.25 − 3.25i)13-s + 1.32i·17-s + (−5.18 + 5.18i)19-s − 4.26i·23-s − 2.20i·25-s + (4.51 − 4.51i)29-s + 5.06i·31-s + (1.18 + 1.18i)35-s + (−3.79 + 3.79i)37-s + 0.186·41-s + (−7.38 − 7.38i)43-s − 4.80·47-s + ⋯
L(s)  = 1  + (0.528 + 0.528i)5-s + 0.377·7-s + (−0.548 + 0.548i)11-s + (−0.902 − 0.902i)13-s + 0.321i·17-s + (−1.18 + 1.18i)19-s − 0.889i·23-s − 0.441i·25-s + (0.837 − 0.837i)29-s + 0.909i·31-s + (0.199 + 0.199i)35-s + (−0.623 + 0.623i)37-s + 0.0291·41-s + (−1.12 − 1.12i)43-s − 0.700·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.877 + 0.479i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1651312769\)
\(L(\frac12)\) \(\approx\) \(0.1651312769\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 1.18i)T + 5iT^{2} \)
11 \( 1 + (1.81 - 1.81i)T - 11iT^{2} \)
13 \( 1 + (3.25 + 3.25i)T + 13iT^{2} \)
17 \( 1 - 1.32iT - 17T^{2} \)
19 \( 1 + (5.18 - 5.18i)T - 19iT^{2} \)
23 \( 1 + 4.26iT - 23T^{2} \)
29 \( 1 + (-4.51 + 4.51i)T - 29iT^{2} \)
31 \( 1 - 5.06iT - 31T^{2} \)
37 \( 1 + (3.79 - 3.79i)T - 37iT^{2} \)
41 \( 1 - 0.186T + 41T^{2} \)
43 \( 1 + (7.38 + 7.38i)T + 43iT^{2} \)
47 \( 1 + 4.80T + 47T^{2} \)
53 \( 1 + (9.62 + 9.62i)T + 53iT^{2} \)
59 \( 1 + (3.57 - 3.57i)T - 59iT^{2} \)
61 \( 1 + (-2.12 - 2.12i)T + 61iT^{2} \)
67 \( 1 + (4.70 - 4.70i)T - 67iT^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + 7.95iT - 73T^{2} \)
79 \( 1 - 1.04iT - 79T^{2} \)
83 \( 1 + (9.07 + 9.07i)T + 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−8.241378105680361622484068776901, −7.42537458218059557439630131316, −6.58640274009686172720681128467, −6.02798519652926622731421772825, −5.05633164296402537384417493470, −4.51318101620110860084568152793, −3.33939014797329875928992366966, −2.45639729489905987345450330702, −1.75563662972657661381991003423, −0.04273329295044741586480117701, 1.42497291274296821405015196264, 2.31140717465646639222262024085, 3.22186000837311624864730340454, 4.52046482753844896113792189859, 4.89432442580872689145665821064, 5.70690211139664482742025361061, 6.58169829869064276810715803106, 7.26567999976686603512388494983, 8.088568505413443052430642166714, 8.823016569172242001546349221422

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