Properties

Degree 2
Conductor $ 2 \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 − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s − 4·13-s − 14-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·21-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s + 28-s − 6·29-s − 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14\)    =    \(2 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 14,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3302236593$
$L(\frac12)$  $\approx$  $0.3302236593$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.58002701984274, −18.21205413166163, −17.21285321618276, −16.33708100139527, −14.60777306567130, −12.30526099005271, −11.23136141438461, −9.765547119459919, −7.575711000888679, −5.579286817429504, 5.579286817429504, 7.575711000888679, 9.765547119459919, 11.23136141438461, 12.30526099005271, 14.60777306567130, 16.33708100139527, 17.21285321618276, 18.21205413166163, 19.58002701984274

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