Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s − 243·3-s + 1.02e3·4-s + 5.76e3·5-s + 7.77e3·6-s + 7.24e4·7-s − 3.27e4·8-s + 5.90e4·9-s − 1.84e5·10-s − 4.08e5·11-s − 2.48e5·12-s + 1.36e6·13-s − 2.31e6·14-s − 1.40e6·15-s + 1.04e6·16-s + 5.42e6·17-s − 1.88e6·18-s + 1.51e7·19-s + 5.90e6·20-s − 1.76e7·21-s + 1.30e7·22-s − 5.21e7·23-s + 7.96e6·24-s − 1.55e7·25-s − 4.37e7·26-s − 1.43e7·27-s + 7.42e7·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.825·5-s + 0.408·6-s + 1.62·7-s − 0.353·8-s + 1/3·9-s − 0.583·10-s − 0.765·11-s − 0.288·12-s + 1.02·13-s − 1.15·14-s − 0.476·15-s + 1/4·16-s + 0.926·17-s − 0.235·18-s + 1.40·19-s + 0.412·20-s − 0.940·21-s + 0.541·22-s − 1.69·23-s + 0.204·24-s − 0.319·25-s − 0.722·26-s − 0.192·27-s + 0.814·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{6} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $1.20558$
$L(\frac12)$  $\approx$  $1.20558$
$L(\frac{13}{2})$   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 \( 1 + p^{5} T \)
3 \( 1 + p^{5} T \)
good5 \( 1 - 5766 T + p^{11} T^{2} \)
7 \( 1 - 10352 p T + p^{11} T^{2} \)
11 \( 1 + 408948 T + p^{11} T^{2} \)
13 \( 1 - 1367558 T + p^{11} T^{2} \)
17 \( 1 - 5422914 T + p^{11} T^{2} \)
19 \( 1 - 15166100 T + p^{11} T^{2} \)
23 \( 1 + 52194072 T + p^{11} T^{2} \)
29 \( 1 - 118581150 T + p^{11} T^{2} \)
31 \( 1 + 57652408 T + p^{11} T^{2} \)
37 \( 1 + 375985186 T + p^{11} T^{2} \)
41 \( 1 - 856316202 T + p^{11} T^{2} \)
43 \( 1 + 1245189172 T + p^{11} T^{2} \)
47 \( 1 + 1306762656 T + p^{11} T^{2} \)
53 \( 1 - 409556358 T + p^{11} T^{2} \)
59 \( 1 + 48862140 p T + p^{11} T^{2} \)
61 \( 1 - 5731767302 T + p^{11} T^{2} \)
67 \( 1 - 3893272244 T + p^{11} T^{2} \)
71 \( 1 + 9075890088 T + p^{11} T^{2} \)
73 \( 1 + 15571822822 T + p^{11} T^{2} \)
79 \( 1 + 30196762600 T + p^{11} T^{2} \)
83 \( 1 - 23135252628 T + p^{11} T^{2} \)
89 \( 1 + 25614819990 T + p^{11} T^{2} \)
97 \( 1 + 61937553406 T + p^{11} T^{2} \)
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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

−20.61408278124063655145673829357, −18.22185823557610957306388607067, −17.74481053097977942632133146030, −16.04683807740066808067323140928, −14.01498326645545396945292831922, −11.66318179051930405686989550799, −10.19821875859845610581844279656, −8.005005940346513327952156957974, −5.55489639023984419351049688700, −1.48555012435943654433178944737, 1.48555012435943654433178944737, 5.55489639023984419351049688700, 8.005005940346513327952156957974, 10.19821875859845610581844279656, 11.66318179051930405686989550799, 14.01498326645545396945292831922, 16.04683807740066808067323140928, 17.74481053097977942632133146030, 18.22185823557610957306388607067, 20.61408278124063655145673829357

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