Properties

Label 2-4032-28.27-c1-0-35
Degree $2$
Conductor $4032$
Sign $0.996 - 0.0839i$
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  + 2.31i·5-s + (−0.222 − 2.63i)7-s + 3.58i·11-s − 2.82i·13-s − 2.31i·17-s − 7.90·19-s + 0.130i·23-s − 0.355·25-s + 0.199·29-s + 3.45·31-s + (6.10 − 0.514i)35-s + 11.5·37-s − 7.97i·41-s + 4.38i·43-s + 6.54·47-s + ⋯
L(s)  = 1  + 1.03i·5-s + (−0.0839 − 0.996i)7-s + 1.08i·11-s − 0.784i·13-s − 0.561i·17-s − 1.81·19-s + 0.0271i·23-s − 0.0711·25-s + 0.0371·29-s + 0.620·31-s + (1.03 − 0.0869i)35-s + 1.90·37-s − 1.24i·41-s + 0.668i·43-s + 0.954·47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.699551935\)
\(L(\frac12)\) \(\approx\) \(1.699551935\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 + 2.63i)T \)
good5 \( 1 - 2.31iT - 5T^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 2.31iT - 17T^{2} \)
19 \( 1 + 7.90T + 19T^{2} \)
23 \( 1 - 0.130iT - 23T^{2} \)
29 \( 1 - 0.199T + 29T^{2} \)
31 \( 1 - 3.45T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 7.97iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 - 6.54T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 6.19iT - 61T^{2} \)
67 \( 1 - 9.01iT - 67T^{2} \)
71 \( 1 - 4.75iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 + 0.768T + 83T^{2} \)
89 \( 1 - 17.7iT - 89T^{2} \)
97 \( 1 - 9.02iT - 97T^{2} \)
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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

−8.264033747045428919674039531255, −7.61062840525673215927126045592, −6.92657147924608983272211889901, −6.53151041557113560498557146080, −5.53525975750279732084273701612, −4.44735483199970370504935049590, −3.98153586145183885195897183357, −2.85685289846444022902768906882, −2.19584967488733657070802555864, −0.70363953043353346940643913160, 0.77606582142321662862091261809, 1.97233472988403162082796243801, 2.80631426209119781256646187835, 4.05593646282766007453009187614, 4.57041715123240760797341306139, 5.59675057360595211343764969040, 6.08323330261651669737783341351, 6.78884388708101202782780464769, 8.071579193991006555500179665687, 8.549803150830717640594735676412

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