Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.605 - 0.795i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3 + 5.19i)11-s + 2·13-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 6·29-s + (−4 + 6.92i)31-s + (−1 − 1.73i)37-s − 12·41-s − 4·43-s + (6 + 10.3i)47-s + (−3 + 5.19i)53-s + (5 + 8.66i)61-s + (−4 + 6.92i)67-s − 6·71-s + ⋯
L(s)  = 1  + (−0.904 + 1.56i)11-s + 0.554·13-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 1.11·29-s + (−0.718 + 1.24i)31-s + (−0.164 − 0.284i)37-s − 1.87·41-s − 0.609·43-s + (0.875 + 1.51i)47-s + (−0.412 + 0.713i)53-s + (0.640 + 1.10i)61-s + (−0.488 + 0.846i)67-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.605 - 0.795i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (361, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.605 - 0.795i)\)
\(L(1)\)  \(\approx\)  \(0.9032890458\)
\(L(\frac12)\)  \(\approx\)  \(0.9032890458\)
\(L(\frac{3}{2})\)   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 \( 1 \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−9.652352784337030265316072572303, −8.753171046328865263242767371009, −7.959904059948904348760877048000, −7.25557579439848056805794805028, −6.45083910272001374724332137794, −5.44433134748165665733168695409, −4.68954446993148147473050923320, −3.76523940903390407862092132855, −2.58461641125149024107019875519, −1.58216474467959829938415842530, 0.32518957930275582294738842578, 1.83445455045527700701481508668, 3.19111518493485473494497950700, 3.71354445295736401136271058640, 5.24181006739380963742293772326, 5.57110160216401925764191329735, 6.61398539613898211369861163130, 7.53772221691430248269179049477, 8.248022322460251768098155017561, 8.964369395771365941610526912509

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