Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $0.947 + 0.318i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.707 − 1.22i)5-s + 0.999·8-s + (0.707 + 1.22i)10-s + 1.41·13-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s − 1.41·20-s + (−0.499 − 0.866i)25-s + (−0.707 + 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)40-s + 1.41·41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.707 − 1.22i)5-s + 0.999·8-s + (0.707 + 1.22i)10-s + 1.41·13-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s − 1.41·20-s + (−0.499 − 0.866i)25-s + (−0.707 + 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)40-s + 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.947 + 0.318i$
motivic weight  =  \(0\)
character  :  $\chi_{1764} (667, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :0),\ 0.947 + 0.318i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9613922268\)
\(L(\frac12)\)  \(\approx\)  \(0.9613922268\)
\(L(1)\)   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 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.229641951583521366417247924755, −8.798183215751483641580790524763, −7.987296331461923099148164036296, −7.10366130523002705997653649497, −6.15735368025110331681855042333, −5.53641733325952376367023765449, −4.82107656251439584939117241219, −3.86118595625707887904445289356, −2.04660180186693198617227803882, −0.947307761203043917555041474569, 1.59659538639705878876861769756, 2.44210338167708534506680586034, 3.50391329763465637679242450302, 4.12641046610760009615627387946, 5.65910093255353624535780162829, 6.35973050890619636035788423990, 7.23483675241046015705311245436, 8.124088053501650816192385652021, 8.939690708279310740518563832350, 9.596853226029443924982534806085

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