Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.468 + 0.883i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.0900i)2-s + (1.98 − 0.254i)4-s − 2.48i·5-s + (2.77 − 0.537i)8-s + (−0.224 − 3.51i)10-s − 4.60·11-s − 5.22·13-s + (3.87 − 1.00i)16-s − 5.61i·17-s − 3.19i·19-s + (−0.632 − 4.93i)20-s + (−6.50 + 0.414i)22-s − 0.718·23-s − 1.19·25-s + (−7.37 + 0.470i)26-s + ⋯
L(s)  = 1  + (0.997 − 0.0636i)2-s + (0.991 − 0.127i)4-s − 1.11i·5-s + (0.981 − 0.189i)8-s + (−0.0708 − 1.11i)10-s − 1.38·11-s − 1.44·13-s + (0.967 − 0.252i)16-s − 1.36i·17-s − 0.732i·19-s + (−0.141 − 1.10i)20-s + (−1.38 + 0.0884i)22-s − 0.149·23-s − 0.238·25-s + (−1.44 + 0.0921i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.468 + 0.883i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1079, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.468 + 0.883i)\)
\(L(1)\)  \(\approx\)  \(2.460200550\)
\(L(\frac12)\)  \(\approx\)  \(2.460200550\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.41 + 0.0900i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.48iT - 5T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 + 5.22T + 13T^{2} \)
17 \( 1 + 5.61iT - 17T^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + 0.718T + 23T^{2} \)
29 \( 1 + 4.53iT - 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 - 2.71T + 37T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 5.41T + 47T^{2} \)
53 \( 1 + 2.06iT - 53T^{2} \)
59 \( 1 + 4.11T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 + 12.6iT - 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 + 9.63T + 73T^{2} \)
79 \( 1 - 8.83iT - 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 8.52iT - 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.194272660591326245070907218664, −7.88810494710679043367759082511, −7.56589333692983975840980321656, −6.52619444625285663762076374938, −5.34554013328489421416692198609, −4.99298138922647577941556539033, −4.36865421408869849289155861155, −2.89350200020861988859114351881, −2.27810031855356558072781439945, −0.59221119386588057815291559640, 2.01911930666568090348988630781, 2.77246687780627039848065499330, 3.59892071685320614296184341369, 4.67135540164652723752187776144, 5.51693656820427724299964278370, 6.22919626299598908772697896779, 7.22214023257848621508649194549, 7.57354577055941463001508670154, 8.521388251158740733806848029307, 10.03651429266043331826788406512

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