Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s − 530·5-s + 120·7-s + 729·9-s − 7.19e3·11-s − 9.62e3·13-s − 1.43e4·15-s + 1.86e4·17-s + 7.00e3·19-s + 3.24e3·21-s − 6.37e4·23-s + 2.02e5·25-s + 1.96e4·27-s + 2.93e4·29-s + 8.79e4·31-s − 1.94e5·33-s − 6.36e4·35-s + 2.27e5·37-s − 2.59e5·39-s − 1.60e5·41-s + 1.36e5·43-s − 3.86e5·45-s − 1.20e6·47-s − 8.09e5·49-s + 5.04e5·51-s − 3.98e5·53-s + 3.81e6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.89·5-s + 0.132·7-s + 1/3·9-s − 1.63·11-s − 1.21·13-s − 1.09·15-s + 0.921·17-s + 0.234·19-s + 0.0763·21-s − 1.09·23-s + 2.59·25-s + 0.192·27-s + 0.223·29-s + 0.530·31-s − 0.941·33-s − 0.250·35-s + 0.739·37-s − 0.701·39-s − 0.364·41-s + 0.261·43-s − 0.632·45-s − 1.69·47-s − 0.982·49-s + 0.532·51-s − 0.367·53-s + 3.09·55-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  =  1
Selberg data  =  $(2,\ 24,\ (\ :7/2),\ -1)$
$L(4)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) 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 \( 1 + 106 p T + p^{7} T^{2} \)
7 \( 1 - 120 T + p^{7} T^{2} \)
11 \( 1 + 7196 T + p^{7} T^{2} \)
13 \( 1 + 9626 T + p^{7} T^{2} \)
17 \( 1 - 18674 T + p^{7} T^{2} \)
19 \( 1 - 7004 T + p^{7} T^{2} \)
23 \( 1 + 63704 T + p^{7} T^{2} \)
29 \( 1 - 29334 T + p^{7} T^{2} \)
31 \( 1 - 87968 T + p^{7} T^{2} \)
37 \( 1 - 227982 T + p^{7} T^{2} \)
41 \( 1 + 160806 T + p^{7} T^{2} \)
43 \( 1 - 136132 T + p^{7} T^{2} \)
47 \( 1 + 25680 p T + p^{7} T^{2} \)
53 \( 1 + 398786 T + p^{7} T^{2} \)
59 \( 1 - 1152436 T + p^{7} T^{2} \)
61 \( 1 + 2070602 T + p^{7} T^{2} \)
67 \( 1 + 4073428 T + p^{7} T^{2} \)
71 \( 1 + 383752 T + p^{7} T^{2} \)
73 \( 1 - 3006010 T + p^{7} T^{2} \)
79 \( 1 + 4948112 T + p^{7} T^{2} \)
83 \( 1 + 9163492 T + p^{7} T^{2} \)
89 \( 1 - 7304106 T + p^{7} T^{2} \)
97 \( 1 + 690526 T + p^{7} T^{2} \)
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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

−15.49634412275897809037816267714, −14.51572267337147330194723114115, −12.75142739404475786718303922868, −11.70158891881792927314915253399, −10.15062334592250504125788176522, −8.105140347838686763686661518023, −7.55996525127021479991118848495, −4.71051886833064557470023330596, −3.04149486888778880849748547661, 0, 3.04149486888778880849748547661, 4.71051886833064557470023330596, 7.55996525127021479991118848495, 8.105140347838686763686661518023, 10.15062334592250504125788176522, 11.70158891881792927314915253399, 12.75142739404475786718303922868, 14.51572267337147330194723114115, 15.49634412275897809037816267714

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