Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s + 11-s + 2·13-s + 2·17-s + 3·19-s + 23-s + 4·25-s − 2·29-s + 5·31-s − 3·35-s + 7·37-s + 7·41-s − 8·43-s − 4·47-s + 49-s − 4·53-s + 3·55-s − 4·59-s + 4·61-s + 6·65-s − 2·67-s − 3·71-s − 77-s − 4·79-s − 2·83-s + 6·85-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s + 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.688·19-s + 0.208·23-s + 4/5·25-s − 0.371·29-s + 0.898·31-s − 0.507·35-s + 1.15·37-s + 1.09·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.549·53-s + 0.404·55-s − 0.520·59-s + 0.512·61-s + 0.744·65-s − 0.244·67-s − 0.356·71-s − 0.113·77-s − 0.450·79-s − 0.219·83-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.869295599$
$L(\frac12)$  $\approx$  $2.869295599$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.081891814446295539844598451747, −7.32731729401683867342261751556, −6.39380538830035948753552261507, −6.07479358668188146111205515708, −5.34392942238886894322572763476, −4.53995044483642793129799651934, −3.49345511979874629830685658071, −2.77357482698646972610902172636, −1.80265320094016597528540454644, −0.941090896084004182569050651738, 0.941090896084004182569050651738, 1.80265320094016597528540454644, 2.77357482698646972610902172636, 3.49345511979874629830685658071, 4.53995044483642793129799651934, 5.34392942238886894322572763476, 6.07479358668188146111205515708, 6.39380538830035948753552261507, 7.32731729401683867342261751556, 8.081891814446295539844598451747

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