Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.998 + 0.0534i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.05 − 0.946i)2-s + (0.207 − 1.98i)4-s − 1.60i·5-s + (−1.66 − 2.28i)8-s + (−1.52 − 1.68i)10-s + 2.67i·11-s − 3.37i·13-s + (−3.91 − 0.823i)16-s − 1.60i·17-s − 4.30·19-s + (−3.19 − 0.333i)20-s + (2.53 + 2.81i)22-s − 6.46i·23-s + 2.41·25-s + (−3.19 − 3.54i)26-s + ⋯
L(s)  = 1  + (0.742 − 0.669i)2-s + (0.103 − 0.994i)4-s − 0.719i·5-s + (−0.588 − 0.808i)8-s + (−0.481 − 0.534i)10-s + 0.807i·11-s − 0.937i·13-s + (−0.978 − 0.205i)16-s − 0.390i·17-s − 0.987·19-s + (−0.715 − 0.0744i)20-s + (0.540 + 0.599i)22-s − 1.34i·23-s + 0.482·25-s + (−0.627 − 0.696i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.998 + 0.0534i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.998 + 0.0534i)\)
\(L(1)\)  \(\approx\)  \(1.881195657\)
\(L(\frac12)\)  \(\approx\)  \(1.881195657\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.05 + 0.946i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.60iT - 5T^{2} \)
11 \( 1 - 2.67iT - 11T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + 1.60iT - 17T^{2} \)
19 \( 1 + 4.30T + 19T^{2} \)
23 \( 1 + 6.46iT - 23T^{2} \)
29 \( 1 - 4.71T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 0.242T + 37T^{2} \)
41 \( 1 + 11.6iT - 41T^{2} \)
43 \( 1 - 7.95iT - 43T^{2} \)
47 \( 1 + 9.04T + 47T^{2} \)
53 \( 1 + 2.46T + 53T^{2} \)
59 \( 1 - 9.04T + 59T^{2} \)
61 \( 1 - 3.37iT - 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 - 8.03iT - 71T^{2} \)
73 \( 1 - 6.17iT - 73T^{2} \)
79 \( 1 + 3.29iT - 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 0.275iT - 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−9.012743990335537815443476512045, −8.304211172496932192925059732920, −7.14804926253507554106939902133, −6.35926894887145491067131184698, −5.33609522187350328319099596813, −4.76942666096787570465839487927, −3.96152323348745887392230972105, −2.82256756250852553413518406581, −1.85529435415175420698527321478, −0.51609221502480383096396745285, 1.95860187502535794551826074907, 3.15956853086506842858357984422, 3.81673782535829819009238969241, 4.82979509922893565702613771704, 5.77223801054791571313272264896, 6.51938054863290119959736528671, 7.04919399465524295611603993182, 7.995095276925466720412160948325, 8.699925381516528639526706895661, 9.501769116568521949988464324433

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