Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.245 - 0.969i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (2.51 − 2.51i)5-s − 7-s + (−0.984 − 0.984i)11-s + (−3.26 + 3.26i)13-s + 5.50i·17-s + (1.21 + 1.21i)19-s + 8.62i·23-s − 7.65i·25-s + (2.04 + 2.04i)29-s + 0.164i·31-s + (−2.51 + 2.51i)35-s + (−8.31 − 8.31i)37-s − 9.56·41-s + (−5.01 + 5.01i)43-s + 3.37·47-s + ⋯
L(s)  = 1  + (1.12 − 1.12i)5-s − 0.377·7-s + (−0.296 − 0.296i)11-s + (−0.904 + 0.904i)13-s + 1.33i·17-s + (0.278 + 0.278i)19-s + 1.79i·23-s − 1.53i·25-s + (0.380 + 0.380i)29-s + 0.0295i·31-s + (−0.425 + 0.425i)35-s + (−1.36 − 1.36i)37-s − 1.49·41-s + (−0.765 + 0.765i)43-s + 0.492·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.245 - 0.969i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.245 - 0.969i)$
$L(1)$  $\approx$  $0.9915626335$
$L(\frac12)$  $\approx$  $0.9915626335$
$L(\frac{3}{2})$   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 \( 1 + (-2.51 + 2.51i)T - 5iT^{2} \)
11 \( 1 + (0.984 + 0.984i)T + 11iT^{2} \)
13 \( 1 + (3.26 - 3.26i)T - 13iT^{2} \)
17 \( 1 - 5.50iT - 17T^{2} \)
19 \( 1 + (-1.21 - 1.21i)T + 19iT^{2} \)
23 \( 1 - 8.62iT - 23T^{2} \)
29 \( 1 + (-2.04 - 2.04i)T + 29iT^{2} \)
31 \( 1 - 0.164iT - 31T^{2} \)
37 \( 1 + (8.31 + 8.31i)T + 37iT^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 + (5.01 - 5.01i)T - 43iT^{2} \)
47 \( 1 - 3.37T + 47T^{2} \)
53 \( 1 + (3.72 - 3.72i)T - 53iT^{2} \)
59 \( 1 + (10.0 + 10.0i)T + 59iT^{2} \)
61 \( 1 + (1.07 - 1.07i)T - 61iT^{2} \)
67 \( 1 + (-3.12 - 3.12i)T + 67iT^{2} \)
71 \( 1 + 7.66iT - 71T^{2} \)
73 \( 1 - 8.40iT - 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + (7.31 - 7.31i)T - 83iT^{2} \)
89 \( 1 + 7.49T + 89T^{2} \)
97 \( 1 + 7.84T + 97T^{2} \)
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

−8.740470646969530908606890297334, −8.080991374592821153887391439287, −7.12304560441935895944917057503, −6.35785520979215511410427051842, −5.51476848602858948441411006167, −5.16944679142148365229257147527, −4.15478356918051932352349579464, −3.23451970343617380898899899636, −1.94656386592670288157645428566, −1.44415539739374706941611405769, 0.25452228738995496738857239892, 1.92442458534532116006600729785, 2.85972217626444811503734420439, 3.07610188055266733360104614901, 4.68847756775547999515981054880, 5.23068211828895463956203198934, 6.12338080094039441274991835924, 6.86010050530083477631703038787, 7.19230743774547151789130270182, 8.216353287002449402828692541465

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