Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.963 + 0.268i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 + 2.12i)5-s − 7-s + (4.03 + 4.03i)11-s + (−4.91 + 4.91i)13-s − 4.76i·17-s + (2.21 + 2.21i)19-s + 6.91i·23-s − 3.99i·25-s + (−2.61 − 2.61i)29-s + 0.712i·31-s + (2.12 − 2.12i)35-s + (5.73 + 5.73i)37-s + 3.59·41-s + (−3.36 + 3.36i)43-s + 6.04·47-s + ⋯
L(s)  = 1  + (−0.948 + 0.948i)5-s − 0.377·7-s + (1.21 + 1.21i)11-s + (−1.36 + 1.36i)13-s − 1.15i·17-s + (0.508 + 0.508i)19-s + 1.44i·23-s − 0.798i·25-s + (−0.485 − 0.485i)29-s + 0.127i·31-s + (0.358 − 0.358i)35-s + (0.943 + 0.943i)37-s + 0.561·41-s + (−0.512 + 0.512i)43-s + 0.881·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.963 + 0.268i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.963 + 0.268i)$
$L(1)$  $\approx$  $0.7625890543$
$L(\frac12)$  $\approx$  $0.7625890543$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + (2.12 - 2.12i)T - 5iT^{2} \)
11 \( 1 + (-4.03 - 4.03i)T + 11iT^{2} \)
13 \( 1 + (4.91 - 4.91i)T - 13iT^{2} \)
17 \( 1 + 4.76iT - 17T^{2} \)
19 \( 1 + (-2.21 - 2.21i)T + 19iT^{2} \)
23 \( 1 - 6.91iT - 23T^{2} \)
29 \( 1 + (2.61 + 2.61i)T + 29iT^{2} \)
31 \( 1 - 0.712iT - 31T^{2} \)
37 \( 1 + (-5.73 - 5.73i)T + 37iT^{2} \)
41 \( 1 - 3.59T + 41T^{2} \)
43 \( 1 + (3.36 - 3.36i)T - 43iT^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + (4.53 - 4.53i)T - 53iT^{2} \)
59 \( 1 + (4.82 + 4.82i)T + 59iT^{2} \)
61 \( 1 + (6.72 - 6.72i)T - 61iT^{2} \)
67 \( 1 + (-3.87 - 3.87i)T + 67iT^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 + 4.61iT - 73T^{2} \)
79 \( 1 - 7.43iT - 79T^{2} \)
83 \( 1 + (-2.44 + 2.44i)T - 83iT^{2} \)
89 \( 1 + 4.23T + 89T^{2} \)
97 \( 1 + 6.76T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.109128032016983132233520648388, −7.62764028693251095074927713015, −7.42485450343007682940283563298, −6.86747176515553714895168631781, −6.12486700734412348352071524954, −4.84551634052185757409791542866, −4.28043822475345523633346428158, −3.48091923205693002021652903953, −2.61347804514321725523833577977, −1.55915545094469068721942793025, 0.26311671685749726060416952298, 1.00151571450968916571004246979, 2.56892768361251018009971368051, 3.52005177339161516400664256500, 4.15099663769928546936367965804, 5.00155470364801613580376242431, 5.78979825127022347412502225941, 6.53907968315827727073821115239, 7.51286107628832465524260563647, 8.070129549440051159040779875199

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