Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
Sign $1$
Motivic weight 6
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s + 286·7-s + 729·9-s + 506·13-s + 1.05e4·19-s + 7.72e3·21-s + 1.56e4·25-s + 1.96e4·27-s − 3.52e4·31-s − 8.92e4·37-s + 1.36e4·39-s − 1.11e5·43-s − 3.58e4·49-s + 2.85e5·57-s − 4.20e5·61-s + 2.08e5·63-s − 1.72e5·67-s + 6.38e5·73-s + 4.21e5·75-s + 2.04e5·79-s + 5.31e5·81-s + 1.44e5·91-s − 9.52e5·93-s − 5.64e4·97-s − 1.12e6·103-s − 2.17e6·109-s − 2.40e6·111-s + ⋯
L(s)  = 1  + 3-s + 0.833·7-s + 9-s + 0.230·13-s + 1.54·19-s + 0.833·21-s + 25-s + 27-s − 1.18·31-s − 1.76·37-s + 0.230·39-s − 1.40·43-s − 0.304·49-s + 1.54·57-s − 1.85·61-s + 0.833·63-s − 0.574·67-s + 1.64·73-s + 75-s + 0.415·79-s + 81-s + 0.192·91-s − 1.18·93-s − 0.0618·97-s − 1.03·103-s − 1.67·109-s − 1.76·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  $\chi_{48} (17, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :3),\ 1)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(2.70429\)
\(L(\frac12)\)  \(\approx\)  \(2.70429\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p^{3} T \)
good5 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
7 \( 1 - 286 T + p^{6} T^{2} \)
11 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
13 \( 1 - 506 T + p^{6} T^{2} \)
17 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
19 \( 1 - 10582 T + p^{6} T^{2} \)
23 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
29 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
31 \( 1 + 35282 T + p^{6} T^{2} \)
37 \( 1 + 89206 T + p^{6} T^{2} \)
41 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
43 \( 1 + 111386 T + p^{6} T^{2} \)
47 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
53 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
59 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
61 \( 1 + 420838 T + p^{6} T^{2} \)
67 \( 1 + 172874 T + p^{6} T^{2} \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( 1 - 638066 T + p^{6} T^{2} \)
79 \( 1 - 204622 T + p^{6} T^{2} \)
83 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
89 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
97 \( 1 + 56446 T + p^{6} T^{2} \)
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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

−14.36036711079201857277652032879, −13.52389063339429235838213761192, −12.16751131503417358404365251918, −10.76564764192639791637036462445, −9.384106802006606075954561307721, −8.270326014845728187440228638173, −7.13427526460697103122579961395, −5.02468006308217887647701318983, −3.33759648271553528484131004379, −1.55590919304050924656819873866, 1.55590919304050924656819873866, 3.33759648271553528484131004379, 5.02468006308217887647701318983, 7.13427526460697103122579961395, 8.270326014845728187440228638173, 9.384106802006606075954561307721, 10.76564764192639791637036462445, 12.16751131503417358404365251918, 13.52389063339429235838213761192, 14.36036711079201857277652032879

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