Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s + 110·5-s + 504·7-s + 729·9-s + 3.81e3·11-s + 9.57e3·13-s + 2.97e3·15-s + 2.60e4·17-s − 3.83e4·19-s + 1.36e4·21-s − 7.11e4·23-s − 6.60e4·25-s + 1.96e4·27-s + 7.42e4·29-s − 2.75e5·31-s + 1.02e5·33-s + 5.54e4·35-s − 2.66e5·37-s + 2.58e5·39-s + 6.84e5·41-s + 2.45e5·43-s + 8.01e4·45-s + 4.78e5·47-s − 5.69e5·49-s + 7.04e5·51-s − 5.69e5·53-s + 4.19e5·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.393·5-s + 0.555·7-s + 1/3·9-s + 0.863·11-s + 1.20·13-s + 0.227·15-s + 1.28·17-s − 1.28·19-s + 0.320·21-s − 1.21·23-s − 0.845·25-s + 0.192·27-s + 0.565·29-s − 1.66·31-s + 0.498·33-s + 0.218·35-s − 0.865·37-s + 0.697·39-s + 1.55·41-s + 0.471·43-s + 0.131·45-s + 0.672·47-s − 0.691·49-s + 0.743·51-s − 0.525·53-s + 0.339·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{24} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 24,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $2.27801$
$L(\frac12)$  $\approx$  $2.27801$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - p^{3} T \)
good5 \( 1 - 22 p T + p^{7} T^{2} \)
7 \( 1 - 72 p T + p^{7} T^{2} \)
11 \( 1 - 3812 T + p^{7} T^{2} \)
13 \( 1 - 9574 T + p^{7} T^{2} \)
17 \( 1 - 26098 T + p^{7} T^{2} \)
19 \( 1 + 38308 T + p^{7} T^{2} \)
23 \( 1 + 71128 T + p^{7} T^{2} \)
29 \( 1 - 74262 T + p^{7} T^{2} \)
31 \( 1 + 275680 T + p^{7} T^{2} \)
37 \( 1 + 266610 T + p^{7} T^{2} \)
41 \( 1 - 684762 T + p^{7} T^{2} \)
43 \( 1 - 245956 T + p^{7} T^{2} \)
47 \( 1 - 478800 T + p^{7} T^{2} \)
53 \( 1 + 569410 T + p^{7} T^{2} \)
59 \( 1 + 1525324 T + p^{7} T^{2} \)
61 \( 1 + 2640458 T + p^{7} T^{2} \)
67 \( 1 - 1416236 T + p^{7} T^{2} \)
71 \( 1 + 3511304 T + p^{7} T^{2} \)
73 \( 1 - 4738618 T + p^{7} T^{2} \)
79 \( 1 - 4661488 T + p^{7} T^{2} \)
83 \( 1 + 5729252 T + p^{7} T^{2} \)
89 \( 1 - 11993514 T + p^{7} T^{2} \)
97 \( 1 - 7150754 T + p^{7} T^{2} \)
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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

−16.09190602319062579992194893959, −14.63221200023629279949614909928, −13.83671489129554663852074268933, −12.33578416001402223357669799575, −10.76228868234596434615313168621, −9.243042176868312796418724021022, −7.953790694673518715589943890035, −6.05317916615898200552832841953, −3.88377140984588093885913230902, −1.66142724699780075595490606581, 1.66142724699780075595490606581, 3.88377140984588093885913230902, 6.05317916615898200552832841953, 7.953790694673518715589943890035, 9.243042176868312796418724021022, 10.76228868234596434615313168621, 12.33578416001402223357669799575, 13.83671489129554663852074268933, 14.63221200023629279949614909928, 16.09190602319062579992194893959

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