Properties

Degree 2
Conductor $ 11 \cdot 17 $
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 + 2·4-s + 4·5-s − 5·7-s − 3·9-s + 8·10-s − 11-s + 4·13-s − 10·14-s − 4·16-s + 17-s − 6·18-s + 2·19-s + 8·20-s − 2·22-s − 2·23-s + 11·25-s + 8·26-s − 10·28-s − 3·29-s + 4·31-s − 8·32-s + 2·34-s − 20·35-s − 6·36-s − 2·37-s + 4·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s − 1.88·7-s − 9-s + 2.52·10-s − 0.301·11-s + 1.10·13-s − 2.67·14-s − 16-s + 0.242·17-s − 1.41·18-s + 0.458·19-s + 1.78·20-s − 0.426·22-s − 0.417·23-s + 11/5·25-s + 1.56·26-s − 1.88·28-s − 0.557·29-s + 0.718·31-s − 1.41·32-s + 0.342·34-s − 3.38·35-s − 36-s − 0.328·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{187} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 187,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.327097641$
$L(\frac12)$  $\approx$  $2.327097641$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−18.76943832319118, −17.95721930807726, −16.86775047518185, −16.13438141026044, −15.13032906751073, −13.85395717099648, −13.66735141164938, −12.96675147622475, −12.05927031115559, −10.64823087958761, −9.656220340411693, −8.878162982617934, −6.617146961127624, −6.011189599885929, −5.423436127881898, −3.502516523512103, −2.608124482023097, 2.608124482023097, 3.502516523512103, 5.423436127881898, 6.011189599885929, 6.617146961127624, 8.878162982617934, 9.656220340411693, 10.64823087958761, 12.05927031115559, 12.96675147622475, 13.66735141164938, 13.85395717099648, 15.13032906751073, 16.13438141026044, 16.86775047518185, 17.95721930807726, 18.76943832319118

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