Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $0.133 - 0.990i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.32i)2-s + (−1.50 + 1.32i)4-s + 1.73i·5-s + (2.50 + 1.32i)8-s + (2.29 − 0.866i)10-s + 2.64i·11-s + 3.46i·13-s + (0.500 − 3.96i)16-s − 6.92i·17-s + (−2.29 − 2.59i)20-s + (3.50 − 1.32i)22-s + 5.29i·23-s + 2.00·25-s + (4.58 − 1.73i)26-s − 5·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.750 + 0.661i)4-s + 0.774i·5-s + (0.883 + 0.467i)8-s + (0.724 − 0.273i)10-s + 0.797i·11-s + 0.960i·13-s + (0.125 − 0.992i)16-s − 1.68i·17-s + (−0.512 − 0.580i)20-s + (0.746 − 0.282i)22-s + 1.10i·23-s + 0.400·25-s + (0.898 − 0.339i)26-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.133 - 0.990i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ 0.133 - 0.990i)\)
\(L(1)\)  \(\approx\)  \(0.8200886341\)
\(L(\frac12)\)  \(\approx\)  \(0.8200886341\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.5 + 1.32i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + 9.16T + 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 7.93iT - 79T^{2} \)
83 \( 1 - 4.58T + 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.487249530717181084564735588106, −9.115330788611697096420605006482, −7.83099142584844082336514456237, −7.29381320173410052865729398359, −6.49331606650418778240929965600, −5.09983209461365445667346049628, −4.41061637432354085562968578428, −3.30426640783988546062337694758, −2.54510329369379419553273193999, −1.46131773783984180374372413399, 0.36522061699548891888520923289, 1.58464797345731596690516060120, 3.32524312924954652001708036831, 4.38328095269047818774587909125, 5.20182501429457144821064245584, 5.99289540515879073945625565177, 6.58356058048873490179702537438, 7.81940322769711816779573288865, 8.312759318337020633138409021045, 8.800870963372520952441849305030

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