Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.224 + 0.974i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.629 − 1.26i)2-s + (−1.20 + 1.59i)4-s + 2.32i·5-s + (2.77 + 0.524i)8-s + (2.94 − 1.46i)10-s − 3.58i·11-s − 2.93i·13-s + (−1.08 − 3.84i)16-s + 2.32i·17-s − 8.33·19-s + (−3.71 − 2.80i)20-s + (−4.53 + 2.25i)22-s − 1.48i·23-s − 0.414·25-s + (−3.71 + 1.84i)26-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (−0.603 + 0.797i)4-s + 1.04i·5-s + (0.982 + 0.185i)8-s + (0.931 − 0.463i)10-s − 1.07i·11-s − 0.812i·13-s + (−0.271 − 0.962i)16-s + 0.564i·17-s − 1.91·19-s + (−0.829 − 0.628i)20-s + (−0.966 + 0.480i)22-s − 0.309i·23-s − 0.0828·25-s + (−0.727 + 0.361i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.224 + 0.974i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.224 + 0.974i)\)
\(L(1)\)  \(\approx\)  \(0.9649550075\)
\(L(\frac12)\)  \(\approx\)  \(0.9649550075\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.629 + 1.26i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.32iT - 5T^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 - 2.32iT - 17T^{2} \)
19 \( 1 + 8.33T + 19T^{2} \)
23 \( 1 + 1.48iT - 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 3.45T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 2.89iT - 41T^{2} \)
43 \( 1 + 6.37iT - 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 8.59T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 2.93iT - 61T^{2} \)
67 \( 1 + 9.02iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + 5.35iT - 97T^{2} \)
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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

−8.915102729936194597775815344463, −8.530233336843525050777465024752, −7.71521424776271505955317973955, −6.70295550891889107743323734547, −5.98894709251861157120270622766, −4.72030363450619593979851421132, −3.68393276887592424095781027015, −2.97721102889748389487957923423, −2.09751363874314606187620955883, −0.48959446679038249125839966977, 1.10363390315352232626702084212, 2.25587598349830082165641903102, 4.27899149507433422377944966363, 4.57449924350644325663591301218, 5.48103120428632071522672456689, 6.57109057056548299159471732682, 7.02379918230400017840373342756, 8.115385599629853228425016420418, 8.681404926187855112876235601458, 9.275162270207830496178264329792

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