Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.577 - 0.816i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s + 2i·5-s − 2.82i·8-s − 2.82·10-s + 7.07·13-s + 4.00·16-s − 2i·17-s − 4.00i·20-s + 25-s + 10.0i·26-s + 9.89i·29-s + 5.65i·32-s + 2.82·34-s + 12·37-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + 0.894i·5-s − 1.00i·8-s − 0.894·10-s + 1.96·13-s + 1.00·16-s − 0.485i·17-s − 0.894i·20-s + 0.200·25-s + 1.96i·26-s + 1.83i·29-s + 1.00i·32-s + 0.485·34-s + 1.97·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.577 - 0.816i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1079, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.577 - 0.816i)\)
\(L(1)\)  \(\approx\)  \(1.643280974\)
\(L(\frac12)\)  \(\approx\)  \(1.643280974\)
\(L(\frac{3}{2})\)   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 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 7.07T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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

−9.155721806403398768800806838786, −8.857355401257650190569389882899, −7.80874456641121904175974254014, −7.17457899748758513362373622160, −6.32956978148163461379656360116, −5.87136029994254922647475097805, −4.75798195623369930991834351556, −3.76150245797625627700967242535, −2.98436293698379732212671002035, −1.17045453004939814686854578233, 0.78552115538202780571156344868, 1.68852368127596592519540870803, 2.99887407347835740493424906111, 4.02452422995861658736526581565, 4.57582245834994345713036511453, 5.72287064771039191265290200714, 6.31623916961709260011527901753, 7.991257448755049527530440139740, 8.304578622662185770140409891221, 9.144720655472036548419316685397

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