Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 223 $
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 + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 5·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s + 18-s − 8·19-s + 20-s + 21-s + 5·22-s + 3·23-s + 24-s + 25-s + 27-s + 28-s + 9·29-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 + 1.50·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.218·21-s + 1.06·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6690\)    =    \(2 \cdot 3 \cdot 5 \cdot 223\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6690} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6690,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.212281001\)
\(L(\frac12)\)  \(\approx\)  \(5.212281001\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;223\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;223\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
223 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−17.28777633773425, −16.59732904024351, −16.08742341253072, −15.10563115846399, −14.78265042689437, −14.42499244006021, −13.75262790139854, −13.10785189566529, −12.66916491080988, −11.96902952739629, −11.20144517538328, −10.77640723088448, −9.962865856724924, −9.129188007079038, −8.753707597351071, −8.035173474641841, −6.988520298821532, −6.625320106644816, −5.956482863284025, −4.998316297492836, −4.190785313672687, −3.867472774395527, −2.643673540599302, −2.070223148758155, −1.098191862754899, 1.098191862754899, 2.070223148758155, 2.643673540599302, 3.867472774395527, 4.190785313672687, 4.998316297492836, 5.956482863284025, 6.625320106644816, 6.988520298821532, 8.035173474641841, 8.753707597351071, 9.129188007079038, 9.962865856724924, 10.77640723088448, 11.20144517538328, 11.96902952739629, 12.66916491080988, 13.10785189566529, 13.75262790139854, 14.42499244006021, 14.78265042689437, 15.10563115846399, 16.08742341253072, 16.59732904024351, 17.28777633773425

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