Properties

Degree $2$
Conductor $1764$
Sign $-0.577 + 0.816i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.643280974\)
\(L(\frac12)\) \(\approx\) \(1.643280974\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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 \( 1 - 18.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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