Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 21-s − 4·22-s − 8·23-s + 24-s + 25-s + 2·26-s − 27-s − 28-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 + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(176610\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{176610} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 176610,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
29 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.73751893996795, −12.96002158607986, −12.65139232573992, −12.11567105329621, −11.78597178064950, −11.24807774360114, −10.78012725269401, −10.15633736093034, −9.996495131187522, −9.414216413950821, −8.856179183151821, −8.650366329021349, −7.841208789859405, −7.324645261685461, −6.876822164674806, −6.337351206569285, −5.969021955011886, −5.636625839903854, −4.644569562414045, −4.287471938961240, −3.761944189993901, −2.897585838395784, −2.357547746238557, −1.614819302621079, −1.299605744807361, 0, 0, 1.299605744807361, 1.614819302621079, 2.357547746238557, 2.897585838395784, 3.761944189993901, 4.287471938961240, 4.644569562414045, 5.636625839903854, 5.969021955011886, 6.337351206569285, 6.876822164674806, 7.324645261685461, 7.841208789859405, 8.650366329021349, 8.856179183151821, 9.414216413950821, 9.996495131187522, 10.15633736093034, 10.78012725269401, 11.24807774360114, 11.78597178064950, 12.11567105329621, 12.65139232573992, 12.96002158607986, 13.73751893996795

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