Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 1051 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 6·19-s − 20-s + 21-s + 22-s + 3·23-s + 24-s − 4·25-s + 27-s + 28-s + 2·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(69366\)    =    \(2 \cdot 3 \cdot 11 \cdot 1051\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{69366} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 69366,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;1051\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;1051\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
1051 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−14.36012208097760, −14.11047504067868, −13.33609936785773, −13.01760209673552, −12.47556851470082, −11.85366008842461, −11.48274962599637, −11.14936219067661, −10.33250706801837, −9.823448812964992, −9.387564127064766, −8.734182032567137, −8.112192147432610, −7.574104596536617, −7.373773200549387, −6.627955193805380, −5.982225730705540, −5.294470124161541, −4.971076922809180, −4.192303777813764, −3.636662088318986, −3.187592980695180, −2.654864602692012, −1.534661553203190, −1.370623397273571, 0, 1.370623397273571, 1.534661553203190, 2.654864602692012, 3.187592980695180, 3.636662088318986, 4.192303777813764, 4.971076922809180, 5.294470124161541, 5.982225730705540, 6.627955193805380, 7.373773200549387, 7.574104596536617, 8.112192147432610, 8.734182032567137, 9.387564127064766, 9.823448812964992, 10.33250706801837, 11.14936219067661, 11.48274962599637, 11.85366008842461, 12.47556851470082, 13.01760209673552, 13.33609936785773, 14.11047504067868, 14.36012208097760

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