Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 81·3-s + 256·4-s + 2.69e3·5-s − 1.29e3·6-s − 3.54e3·7-s − 4.09e3·8-s + 6.56e3·9-s − 4.31e4·10-s + 2.95e4·11-s + 2.07e4·12-s − 4.48e4·13-s + 5.67e4·14-s + 2.18e5·15-s + 6.55e4·16-s − 1.01e5·17-s − 1.04e5·18-s − 8.95e5·19-s + 6.89e5·20-s − 2.87e5·21-s − 4.73e5·22-s − 1.11e6·23-s − 3.31e5·24-s + 5.30e6·25-s + 7.17e5·26-s + 5.31e5·27-s − 9.07e5·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.92·5-s − 0.408·6-s − 0.557·7-s − 0.353·8-s + 1/3·9-s − 1.36·10-s + 0.609·11-s + 0.288·12-s − 0.435·13-s + 0.394·14-s + 1.11·15-s + 1/4·16-s − 0.296·17-s − 0.235·18-s − 1.57·19-s + 0.963·20-s − 0.322·21-s − 0.430·22-s − 0.829·23-s − 0.204·24-s + 2.71·25-s + 0.307·26-s + 0.192·27-s − 0.278·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  $\chi_{6} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :9/2),\ 1)$
$L(5)$  $\approx$  $1.44504$
$L(\frac12)$  $\approx$  $1.44504$
$L(\frac{11}{2})$   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 \( 1 + p^{4} T \)
3 \( 1 - p^{4} T \)
good5 \( 1 - 2694 T + p^{9} T^{2} \)
7 \( 1 + 3544 T + p^{9} T^{2} \)
11 \( 1 - 29580 T + p^{9} T^{2} \)
13 \( 1 + 44818 T + p^{9} T^{2} \)
17 \( 1 + 101934 T + p^{9} T^{2} \)
19 \( 1 + 895084 T + p^{9} T^{2} \)
23 \( 1 + 1113000 T + p^{9} T^{2} \)
29 \( 1 + 2357346 T + p^{9} T^{2} \)
31 \( 1 - 175808 T + p^{9} T^{2} \)
37 \( 1 + 2919418 T + p^{9} T^{2} \)
41 \( 1 - 26218794 T + p^{9} T^{2} \)
43 \( 1 + 436348 p T + p^{9} T^{2} \)
47 \( 1 + 20966160 T + p^{9} T^{2} \)
53 \( 1 - 57251574 T + p^{9} T^{2} \)
59 \( 1 - 33587580 T + p^{9} T^{2} \)
61 \( 1 - 82260830 T + p^{9} T^{2} \)
67 \( 1 + 188455804 T + p^{9} T^{2} \)
71 \( 1 - 80924040 T + p^{9} T^{2} \)
73 \( 1 + 236140918 T + p^{9} T^{2} \)
79 \( 1 - 526909808 T + p^{9} T^{2} \)
83 \( 1 - 18346452 T + p^{9} T^{2} \)
89 \( 1 - 690643098 T + p^{9} T^{2} \)
97 \( 1 + 438251038 T + p^{9} T^{2} \)
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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

−20.96156601394129440812055787588, −19.44620417038448737645742863102, −17.90754019954750947617402377460, −16.75010533851616041963519763457, −14.56014998208743926737478615457, −13.03935611020553074998403169273, −10.17863203967913836258984678197, −9.060136005069787656827173382478, −6.38299705502718747870187673783, −2.09518556355666538401470846280, 2.09518556355666538401470846280, 6.38299705502718747870187673783, 9.060136005069787656827173382478, 10.17863203967913836258984678197, 13.03935611020553074998403169273, 14.56014998208743926737478615457, 16.75010533851616041963519763457, 17.90754019954750947617402377460, 19.44620417038448737645742863102, 20.96156601394129440812055787588

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