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  + 1.79·5-s − 7-s + 2.96·11-s + 6.95·13-s − 6.11·17-s + 0.967·19-s + 7.91·23-s − 1.76·25-s + 0.798·29-s + 0.798·31-s − 1.79·35-s + 6.76·37-s + 3.79·41-s − 9.71·43-s − 11.0·47-s + 49-s + 13.6·53-s + 5.33·55-s + 4.59·59-s + 8.56·61-s + 12.4·65-s + 13.5·67-s − 11.9·71-s + 4.33·73-s − 2.96·77-s − 17.6·79-s − 0.765·83-s + ⋯
L(s)  = 1  + 0.804·5-s − 0.377·7-s + 0.894·11-s + 1.92·13-s − 1.48·17-s + 0.221·19-s + 1.65·23-s − 0.353·25-s + 0.148·29-s + 0.143·31-s − 0.303·35-s + 1.11·37-s + 0.593·41-s − 1.48·43-s − 1.61·47-s + 0.142·49-s + 1.87·53-s + 0.719·55-s + 0.598·59-s + 1.09·61-s + 1.55·65-s + 1.65·67-s − 1.41·71-s + 0.507·73-s − 0.338·77-s − 1.98·79-s − 0.0840·83-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.706236204$
$L(\frac12)$  $\approx$  $2.706236204$
$L(\frac{3}{2})$   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 \( 1 - 1.79T + 5T^{2} \)
11 \( 1 - 2.96T + 11T^{2} \)
13 \( 1 - 6.95T + 13T^{2} \)
17 \( 1 + 6.11T + 17T^{2} \)
19 \( 1 - 0.967T + 19T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 - 0.798T + 29T^{2} \)
31 \( 1 - 0.798T + 31T^{2} \)
37 \( 1 - 6.76T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 + 9.71T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 4.59T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 4.33T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 0.765T + 83T^{2} \)
89 \( 1 - 3.98T + 89T^{2} \)
97 \( 1 + 6.86T + 97T^{2} \)
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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

−8.472006266376794796094313907574, −7.05974819667598045051562242489, −6.61305770801808490183361955181, −6.07272988745430721204638771924, −5.34178357917048277128398504645, −4.33812409878164773533152275671, −3.67400830954429388307782575039, −2.78221419890145217350491935060, −1.75423328404891786128253559311, −0.922085834954030413294704919068, 0.922085834954030413294704919068, 1.75423328404891786128253559311, 2.78221419890145217350491935060, 3.67400830954429388307782575039, 4.33812409878164773533152275671, 5.34178357917048277128398504645, 6.07272988745430721204638771924, 6.61305770801808490183361955181, 7.05974819667598045051562242489, 8.472006266376794796094313907574

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