Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.456 + 0.889i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.925 + 0.925i)5-s − 7-s + (−1.72 − 1.72i)11-s + (0.328 − 0.328i)13-s + 2.34i·17-s + (1.77 + 1.77i)19-s − 6.17i·23-s + 3.28i·25-s + (0.122 + 0.122i)29-s + 1.74i·31-s + (0.925 − 0.925i)35-s + (−1.68 − 1.68i)37-s − 2.88·41-s + (−2.77 + 2.77i)43-s + 5.92·47-s + ⋯
L(s)  = 1  + (−0.413 + 0.413i)5-s − 0.377·7-s + (−0.519 − 0.519i)11-s + (0.0910 − 0.0910i)13-s + 0.567i·17-s + (0.408 + 0.408i)19-s − 1.28i·23-s + 0.657i·25-s + (0.0227 + 0.0227i)29-s + 0.313i·31-s + (0.156 − 0.156i)35-s + (−0.276 − 0.276i)37-s − 0.451·41-s + (−0.422 + 0.422i)43-s + 0.863·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.456 + 0.889i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.456 + 0.889i)$
$L(1)$  $\approx$  $1.095343602$
$L(\frac12)$  $\approx$  $1.095343602$
$L(\frac{3}{2})$   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 \( 1 + (0.925 - 0.925i)T - 5iT^{2} \)
11 \( 1 + (1.72 + 1.72i)T + 11iT^{2} \)
13 \( 1 + (-0.328 + 0.328i)T - 13iT^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
19 \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \)
23 \( 1 + 6.17iT - 23T^{2} \)
29 \( 1 + (-0.122 - 0.122i)T + 29iT^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (1.68 + 1.68i)T + 37iT^{2} \)
41 \( 1 + 2.88T + 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 - 5.92T + 47T^{2} \)
53 \( 1 + (-0.973 + 0.973i)T - 53iT^{2} \)
59 \( 1 + (8.33 + 8.33i)T + 59iT^{2} \)
61 \( 1 + (-4.28 + 4.28i)T - 61iT^{2} \)
67 \( 1 + (1.78 + 1.78i)T + 67iT^{2} \)
71 \( 1 + 8.57iT - 71T^{2} \)
73 \( 1 + 6.41iT - 73T^{2} \)
79 \( 1 - 5.38iT - 79T^{2} \)
83 \( 1 + (3.46 - 3.46i)T - 83iT^{2} \)
89 \( 1 - 1.51T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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

−8.258679742890440225299549991523, −7.63980139773891728036235134130, −6.83103375258587685795307736425, −6.16425183094287576634562215009, −5.39522380284727654051657525676, −4.49313415183184641640879282628, −3.50716079167165528366342249811, −3.00484682531933031535988549343, −1.82740014960597603546727538636, −0.38829300399291091391476745898, 0.911424531986132050798148823331, 2.20101526223365760319930356302, 3.13706846273051825200657031041, 4.02150285165938446986112274756, 4.84085991779562362013044504600, 5.49694716085821523324045884347, 6.39098573124297855701697317599, 7.33221008111893547621695320005, 7.64768175467788615355060336690, 8.671556511657917890099776650292

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