Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \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 − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 11-s − 12-s + 4·13-s − 2·14-s + 16-s + 6·17-s + 18-s − 4·19-s + 2·21-s − 22-s − 6·23-s − 24-s + 4·26-s − 27-s − 2·28-s + 6·29-s + 8·31-s + 32-s + 33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.174·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1650} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1650,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.138820819$
$L(\frac12)$  $\approx$  $2.138820819$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−19.75193466233010, −19.18418444054954, −18.48202136291421, −17.84121963689530, −16.93770735722919, −16.30649336190092, −15.91406278918153, −15.19779381743298, −14.32663437286031, −13.65050774601895, −12.99933297565399, −12.33973798850639, −11.78782375087704, −10.93044849785759, −10.20523186462138, −9.680870325678190, −8.354832042825880, −7.781831145868319, −6.433421528398456, −6.282978262839684, −5.367237323607380, −4.330197124097081, −3.558747604712981, −2.505551770882837, −0.9741752983756780, 0.9741752983756780, 2.505551770882837, 3.558747604712981, 4.330197124097081, 5.367237323607380, 6.282978262839684, 6.433421528398456, 7.781831145868319, 8.354832042825880, 9.680870325678190, 10.20523186462138, 10.93044849785759, 11.78782375087704, 12.33973798850639, 12.99933297565399, 13.65050774601895, 14.32663437286031, 15.19779381743298, 15.91406278918153, 16.30649336190092, 16.93770735722919, 17.84121963689530, 18.48202136291421, 19.18418444054954, 19.75193466233010

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