Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 11 $
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 + 4-s + 2·5-s − 7-s + 8-s − 3·9-s + 2·10-s − 11-s + 2·13-s − 14-s + 16-s + 2·17-s − 3·18-s + 2·20-s − 22-s − 8·23-s − 25-s + 2·26-s − 28-s − 2·29-s − 8·31-s + 32-s + 2·34-s − 2·35-s − 3·36-s − 2·37-s + 2·40-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.447·20-s − 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.338·35-s − 1/2·36-s − 0.328·37-s + 0.316·40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(154\)    =    \(2 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{154} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 154,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.689588035$
$L(\frac12)$  $\approx$  $1.689588035$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−19.54606788612343, −18.34995373195942, −17.52889711962058, −16.55477397890272, −15.79553155219393, −14.52074011948210, −13.95416200626634, −13.10274413461999, −12.12455900124815, −11.04890213928127, −10.04454810525505, −8.919520401542258, −7.581874794264126, −6.026435913010643, −5.606179608230283, −3.784763852357839, −2.309843635216443, 2.309843635216443, 3.784763852357839, 5.606179608230283, 6.026435913010643, 7.581874794264126, 8.919520401542258, 10.04454810525505, 11.04890213928127, 12.12455900124815, 13.10274413461999, 13.95416200626634, 14.52074011948210, 15.79553155219393, 16.55477397890272, 17.52889711962058, 18.34995373195942, 19.54606788612343

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