↑

Properties

Label 2-83391-1.1-c1-0-9
Degree $2$
Conductor $83391$
Sign $1$
Analytic cond. $665.880$
Root an. cond. $25.8046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 3·5-s + 2·6-s + 7-s + 9-s + 6·10-s − 11-s + 2·12-s − 4·13-s + 2·14-s + 3·15-s − 4·16-s + 2·18-s + 6·20-s + 21-s − 2·22-s + 2·23-s + 4·25-s − 8·26-s + 27-s + 2·28-s + 8·29-s + 6·30-s − 5·31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 1.34·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.89·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s + 0.534·14-s + 0.774·15-s − 16-s + 0.471·18-s + 1.34·20-s + 0.218·21-s − 0.426·22-s + 0.417·23-s + 4/5·25-s − 1.56·26-s + 0.192·27-s + 0.377·28-s + 1.48·29-s + 1.09·30-s − 0.898·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(83391\)    =    \(3 \cdot 7 \cdot 11 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(665.880\)
Root analytic conductor: \(25.8046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 83391,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.556582482\)
\(L(\frac12)\) \(\approx\) \(9.556582482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.87439273067621, −13.66524437331018, −13.01577938640115, −12.66864653290334, −12.22847370646740, −11.72120013300317, −10.88616514160621, −10.65706775581557, −9.777194351108443, −9.553955449059472, −9.074974231694384, −8.319700752087821, −7.854080718041946, −6.976333055450656, −6.783558080965066, −6.016490389479651, −5.498611252871122, −5.147432812197290, −4.544551444392571, −4.096808959321412, −3.244360126837136, −2.632900135200476, −2.363203096882942, −1.710448963777328, −0.7201165866051855, 0.7201165866051855, 1.710448963777328, 2.363203096882942, 2.632900135200476, 3.244360126837136, 4.096808959321412, 4.544551444392571, 5.147432812197290, 5.498611252871122, 6.016490389479651, 6.783558080965066, 6.976333055450656, 7.854080718041946, 8.319700752087821, 9.074974231694384, 9.553955449059472, 9.777194351108443, 10.65706775581557, 10.88616514160621, 11.72120013300317, 12.22847370646740, 12.66864653290334, 13.01577938640115, 13.66524437331018, 13.87439273067621

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