Properties

Degree 2
Conductor $ 2 \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  + 8·2-s + 27·3-s + 64·4-s − 114·5-s + 216·6-s − 1.57e3·7-s + 512·8-s + 729·9-s − 912·10-s + 7.33e3·11-s + 1.72e3·12-s − 3.80e3·13-s − 1.26e4·14-s − 3.07e3·15-s + 4.09e3·16-s − 6.60e3·17-s + 5.83e3·18-s + 2.48e4·19-s − 7.29e3·20-s − 4.25e4·21-s + 5.86e4·22-s + 4.14e4·23-s + 1.38e4·24-s − 6.51e4·25-s − 3.04e4·26-s + 1.96e4·27-s − 1.00e5·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.407·5-s + 0.408·6-s − 1.73·7-s + 0.353·8-s + 1/3·9-s − 0.288·10-s + 1.66·11-s + 0.288·12-s − 0.479·13-s − 1.22·14-s − 0.235·15-s + 1/4·16-s − 0.326·17-s + 0.235·18-s + 0.831·19-s − 0.203·20-s − 1.00·21-s + 1.17·22-s + 0.710·23-s + 0.204·24-s − 0.833·25-s − 0.339·26-s + 0.192·27-s − 0.868·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{6} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $1.77288$
$L(\frac12)$  $\approx$  $1.77288$
$L(\frac{9}{2})$   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 \( 1 - p^{3} T \)
3 \( 1 - p^{3} T \)
good5 \( 1 + 114 T + p^{7} T^{2} \)
7 \( 1 + 1576 T + p^{7} T^{2} \)
11 \( 1 - 7332 T + p^{7} T^{2} \)
13 \( 1 + 3802 T + p^{7} T^{2} \)
17 \( 1 + 6606 T + p^{7} T^{2} \)
19 \( 1 - 24860 T + p^{7} T^{2} \)
23 \( 1 - 41448 T + p^{7} T^{2} \)
29 \( 1 + 41610 T + p^{7} T^{2} \)
31 \( 1 - 33152 T + p^{7} T^{2} \)
37 \( 1 + 36466 T + p^{7} T^{2} \)
41 \( 1 + 639078 T + p^{7} T^{2} \)
43 \( 1 + 156412 T + p^{7} T^{2} \)
47 \( 1 + 433776 T + p^{7} T^{2} \)
53 \( 1 - 786078 T + p^{7} T^{2} \)
59 \( 1 - 745140 T + p^{7} T^{2} \)
61 \( 1 + 1660618 T + p^{7} T^{2} \)
67 \( 1 + 3290836 T + p^{7} T^{2} \)
71 \( 1 - 5716152 T + p^{7} T^{2} \)
73 \( 1 - 2659898 T + p^{7} T^{2} \)
79 \( 1 - 3807440 T + p^{7} T^{2} \)
83 \( 1 - 2229468 T + p^{7} T^{2} \)
89 \( 1 - 5991210 T + p^{7} T^{2} \)
97 \( 1 + 4060126 T + p^{7} T^{2} \)
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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

−21.99212808302799220287334832597, −19.98516930348493286017205517031, −19.29694332381488804409734755114, −16.61276172094097152096802371309, −15.21195878924197135927898708521, −13.58662154545969536227601741403, −12.06958542760472821610344422273, −9.500006282710758553856968294892, −6.76806031106170880515924959467, −3.53997220835025805300938543853, 3.53997220835025805300938543853, 6.76806031106170880515924959467, 9.500006282710758553856968294892, 12.06958542760472821610344422273, 13.58662154545969536227601741403, 15.21195878924197135927898708521, 16.61276172094097152096802371309, 19.29694332381488804409734755114, 19.98516930348493286017205517031, 21.99212808302799220287334832597

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