Properties

Label 2-4032-56.27-c1-0-72
Degree $2$
Conductor $4032$
Sign $-0.436 + 0.899i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 2i)7-s + 2·13-s − 8i·19-s − 5·25-s − 10.3·31-s − 6.92i·37-s + 10.3·43-s + (−1.00 − 6.92i)49-s − 14·61-s + 3.46·67-s + 13.8i·73-s − 4i·79-s + (3.46 − 4i)91-s + 13.8i·97-s + 3.46·103-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)7-s + 0.554·13-s − 1.83i·19-s − 25-s − 1.86·31-s − 1.13i·37-s + 1.58·43-s + (−0.142 − 0.989i)49-s − 1.79·61-s + 0.423·67-s + 1.62i·73-s − 0.450i·79-s + (0.363 − 0.419i)91-s + 1.40i·97-s + 0.341·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.436 + 0.899i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.467447006\)
\(L(\frac12)\) \(\approx\) \(1.467447006\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.121027130941576072789545312537, −7.41349557237542373473196166129, −6.95128665535152032641806836758, −5.92037058083739490436620535020, −5.21772892173754901226165429290, −4.31632737346917584689058785434, −3.74379319773414750979862277512, −2.58968125582364013823318101683, −1.58708634058047786718257389312, −0.41307889716314904907502465453, 1.42327199973411291946639229461, 2.12582251292371223293178145075, 3.33328704947412590409297021598, 4.05489222235286947751049286689, 5.01932708388041304516580445284, 5.85786076859312040007306860622, 6.16895249003112138026847430187, 7.49194387697765393318712401387, 7.87763342728104376916579699839, 8.677581504196959099947495339732

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