Properties

Label 2-4032-24.11-c1-0-41
Degree $2$
Conductor $4032$
Sign $-0.169 + 0.985i$
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  + 2.82·5-s i·7-s − 1.41i·11-s − 2i·13-s − 2.82i·17-s − 4·19-s − 1.41·23-s + 3.00·25-s + 1.41·29-s − 2.82i·35-s − 10i·37-s − 5.65i·41-s − 2·43-s − 2.82·47-s − 49-s + ⋯
L(s)  = 1  + 1.26·5-s − 0.377i·7-s − 0.426i·11-s − 0.554i·13-s − 0.685i·17-s − 0.917·19-s − 0.294·23-s + 0.600·25-s + 0.262·29-s − 0.478i·35-s − 1.64i·37-s − 0.883i·41-s − 0.304·43-s − 0.412·47-s − 0.142·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.169 + 0.985i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.884484792\)
\(L(\frac12)\) \(\approx\) \(1.884484792\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 1.41T + 53T^{2} \)
59 \( 1 + 8.48iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.309064096979792902471737894869, −7.46076956882356859666521733744, −6.69427386633890933464922714377, −5.94220038570519448194916089165, −5.43257936317300005251271268971, −4.52248975144401655074347346632, −3.54786167550386909182670310327, −2.56013378497228691770130318864, −1.77698228163816288316211788801, −0.49595681400993365209837501461, 1.47887309262312143815719468700, 2.11047854104509655797735025317, 3.02552300158137984709908375922, 4.22706873670196751718779378085, 4.91713313088572057797105714482, 5.84872182338975117187854715848, 6.33841804191066238365394184724, 6.96748049986969442083375271466, 8.107402387682625284817163465704, 8.635183100566230240293227349718

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