Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s − 2·5-s + 2·6-s + 9-s − 4·10-s − 2·11-s + 2·12-s + 13-s − 2·15-s − 4·16-s + 2·18-s + 19-s − 4·20-s − 4·22-s − 25-s + 2·26-s + 27-s + 4·29-s − 4·30-s + 9·31-s − 8·32-s − 2·33-s + 2·36-s + 3·37-s + 2·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s + 1/3·9-s − 1.26·10-s − 0.603·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s + 0.229·19-s − 0.894·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.742·29-s − 0.730·30-s + 1.61·31-s − 1.41·32-s − 0.348·33-s + 1/3·36-s + 0.493·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(147\)    =    \(3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{147} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 147,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.134033734$
$L(\frac12)$  $\approx$  $2.134033734$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
7 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.69223246884804, −18.80396305179846, −17.78261405524769, −16.23018668419908, −15.51938495151802, −14.94403599066949, −13.86466080107451, −13.27367561777592, −12.20598219965914, −11.52656811430916, −10.18473039565837, −8.693773060427938, −7.675838452128204, −6.391993423583046, −4.964559952612693, −3.927560519584507, −2.815020843626137, 2.815020843626137, 3.927560519584507, 4.964559952612693, 6.391993423583046, 7.675838452128204, 8.693773060427938, 10.18473039565837, 11.52656811430916, 12.20598219965914, 13.27367561777592, 13.86466080107451, 14.94403599066949, 15.51938495151802, 16.23018668419908, 17.78261405524769, 18.80396305179846, 19.69223246884804

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