Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4-s − 18·5-s + 9·6-s + 7·7-s + 21·8-s + 9·9-s + 54·10-s − 36·11-s − 3·12-s − 34·13-s − 21·14-s + 54·15-s − 71·16-s + 42·17-s − 27·18-s − 124·19-s − 18·20-s − 21·21-s + 108·22-s − 63·24-s + 199·25-s + 102·26-s − 27·27-s + 7·28-s + 102·29-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.577·3-s + 1/8·4-s − 1.60·5-s + 0.612·6-s + 0.377·7-s + 0.928·8-s + 1/3·9-s + 1.70·10-s − 0.986·11-s − 0.0721·12-s − 0.725·13-s − 0.400·14-s + 0.929·15-s − 1.10·16-s + 0.599·17-s − 0.353·18-s − 1.49·19-s − 0.201·20-s − 0.218·21-s + 1.04·22-s − 0.535·24-s + 1.59·25-s + 0.769·26-s − 0.192·27-s + 0.0472·28-s + 0.653·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(3\)
character  :  $\chi_{21} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 21,\ (\ :3/2),\ -1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + p T \)
7 \( 1 - p T \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
5 \( 1 + 18 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 + 34 T + p^{3} T^{2} \)
17 \( 1 - 42 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 - 102 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 - 398 T + p^{3} T^{2} \)
41 \( 1 + 318 T + p^{3} T^{2} \)
43 \( 1 + 268 T + p^{3} T^{2} \)
47 \( 1 - 240 T + p^{3} T^{2} \)
53 \( 1 + 498 T + p^{3} T^{2} \)
59 \( 1 + 132 T + p^{3} T^{2} \)
61 \( 1 - 398 T + p^{3} T^{2} \)
67 \( 1 - 92 T + p^{3} T^{2} \)
71 \( 1 + 720 T + p^{3} T^{2} \)
73 \( 1 + 502 T + p^{3} T^{2} \)
79 \( 1 + 1024 T + p^{3} T^{2} \)
83 \( 1 + 204 T + p^{3} T^{2} \)
89 \( 1 - 354 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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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

−17.11626101801103868663808827808, −16.14418605050144376207476547610, −14.91290151933212586147181506316, −12.75890770904133744320234829033, −11.38988080428046421269273778849, −10.29314165331111143855582291138, −8.366976436165780451630990829562, −7.43205058596234115633185970014, −4.59505455463720251261035786388, 0, 4.59505455463720251261035786388, 7.43205058596234115633185970014, 8.366976436165780451630990829562, 10.29314165331111143855582291138, 11.38988080428046421269273778849, 12.75890770904133744320234829033, 14.91290151933212586147181506316, 16.14418605050144376207476547610, 17.11626101801103868663808827808

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