Properties

Label 2-4032-12.11-c1-0-11
Degree $2$
Conductor $4032$
Sign $0.577 - 0.816i$
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 − 4.24·11-s + 2·13-s + 2.82i·17-s − 4i·19-s − 1.41·23-s + 5·25-s + 7.07i·29-s + 8i·31-s − 8·37-s − 5.65i·41-s − 4i·43-s + 2.82·47-s − 49-s + 7.07i·53-s + ⋯
L(s)  = 1  − 0.377i·7-s − 1.27·11-s + 0.554·13-s + 0.685i·17-s − 0.917i·19-s − 0.294·23-s + 25-s + 1.31i·29-s + 1.43i·31-s − 1.31·37-s − 0.883i·41-s − 0.609i·43-s + 0.412·47-s − 0.142·49-s + 0.971i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418415064\)
\(L(\frac12)\) \(\approx\) \(1.418415064\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.648971805428006050617047375060, −7.83139131680750401553014403654, −7.02924363304841303261418784468, −6.53250937703357502980314475950, −5.30844715111272461731754665859, −5.05722061423159098304405762604, −3.87116340803381978732974183137, −3.14456048093976067517817961885, −2.15795225530334895685131655374, −0.956525192328075346381689201982, 0.48114363465582557680818607695, 1.94542266879150701101708026921, 2.75650701594980597046193598101, 3.65365226636829543038216837041, 4.63989637901656283314016243023, 5.39946766422453506327710266592, 6.03159415370751041319645720500, 6.84494350768207595297882132723, 7.85225452523527299161281492288, 8.135409885858276448790410107266

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