Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s − 7-s + 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 6·13-s − 14-s + 2·15-s + 16-s + 2·17-s + 18-s − 4·19-s − 2·20-s + 21-s − 4·22-s + 8·23-s − 24-s − 25-s + 6·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(42\)    =    \(2 \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{42} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 42,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8688618643$
$L(\frac12)$  $\approx$  $0.8688618643$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.24069862988852, −18.44852109750560, −16.87411793531398, −15.81961508686076, −15.24725628375398, −13.48218593690235, −12.65213238808597, −11.37318927677395, −10.51951977961492, −8.356918966632092, −6.845524806029865, −5.339850147876028, −3.624828870818865, 3.624828870818865, 5.339850147876028, 6.845524806029865, 8.356918966632092, 10.51951977961492, 11.37318927677395, 12.65213238808597, 13.48218593690235, 15.24725628375398, 15.81961508686076, 16.87411793531398, 18.44852109750560, 19.24069862988852

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