Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s + 6·5-s + 6·6-s − 16·7-s − 8·8-s + 9·9-s − 12·10-s + 12·11-s − 12·12-s + 38·13-s + 32·14-s − 18·15-s + 16·16-s − 126·17-s − 18·18-s + 20·19-s + 24·20-s + 48·21-s − 24·22-s + 168·23-s + 24·24-s − 89·25-s − 76·26-s − 27·27-s − 64·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.536·5-s + 0.408·6-s − 0.863·7-s − 0.353·8-s + 1/3·9-s − 0.379·10-s + 0.328·11-s − 0.288·12-s + 0.810·13-s + 0.610·14-s − 0.309·15-s + 1/4·16-s − 1.79·17-s − 0.235·18-s + 0.241·19-s + 0.268·20-s + 0.498·21-s − 0.232·22-s + 1.52·23-s + 0.204·24-s − 0.711·25-s − 0.573·26-s − 0.192·27-s − 0.431·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{6} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $0.509710$
$L(\frac12)$  $\approx$  $0.509710$
$L(\frac{5}{2})$   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 \( 1 + p T \)
3 \( 1 + p T \)
good5 \( 1 - 6 T + p^{3} T^{2} \)
7 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 126 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 - 168 T + p^{3} T^{2} \)
29 \( 1 - 30 T + p^{3} T^{2} \)
31 \( 1 + 88 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 - 42 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 + 96 T + p^{3} T^{2} \)
53 \( 1 - 198 T + p^{3} T^{2} \)
59 \( 1 + 660 T + p^{3} T^{2} \)
61 \( 1 + 538 T + p^{3} T^{2} \)
67 \( 1 - 884 T + p^{3} T^{2} \)
71 \( 1 - 792 T + p^{3} T^{2} \)
73 \( 1 - 218 T + p^{3} T^{2} \)
79 \( 1 + 520 T + p^{3} T^{2} \)
83 \( 1 + 492 T + p^{3} T^{2} \)
89 \( 1 - 810 T + p^{3} T^{2} \)
97 \( 1 - 1154 T + p^{3} T^{2} \)
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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

−22.97901126778615505977980355311, −21.58778001713410487232827259598, −19.86669072742657907925171947529, −18.33094748384698500881885833386, −17.06455999699601621437849501998, −15.67334368036257763172445032023, −13.15736614711876148220321478103, −11.07166341738426389307147212093, −9.298914371498593683037799278765, −6.48044763846723198372779155495, 6.48044763846723198372779155495, 9.298914371498593683037799278765, 11.07166341738426389307147212093, 13.15736614711876148220321478103, 15.67334368036257763172445032023, 17.06455999699601621437849501998, 18.33094748384698500881885833386, 19.86669072742657907925171947529, 21.58778001713410487232827259598, 22.97901126778615505977980355311

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