Properties

Label 2-327990-1.1-c1-0-42
Degree $2$
Conductor $327990$
Sign $-1$
Analytic cond. $2619.01$
Root an. cond. $51.1762$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 2·11-s + 12-s + 13-s + 14-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s + 21-s + 2·22-s − 23-s + 24-s + 25-s + 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 + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(327990\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2619.01\)
Root analytic conductor: \(51.1762\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
29 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−13.07994857091434, −12.44651132659504, −11.86319723384639, −11.59766378308502, −11.09615813904766, −10.66214422993380, −10.07371920024351, −9.530961833070760, −9.302342079864233, −8.693793154101124, −8.229692095606365, −7.677040771188120, −7.275729373268207, −6.736491630145557, −6.249432931750399, −5.853415222997593, −5.143286557068877, −4.810340235631026, −4.326948145587147, −3.567233098661925, −3.329302105637843, −2.763631998607047, −1.952777831879319, −1.664019257579570, −1.093947547081615, 0, 1.093947547081615, 1.664019257579570, 1.952777831879319, 2.763631998607047, 3.329302105637843, 3.567233098661925, 4.326948145587147, 4.810340235631026, 5.143286557068877, 5.853415222997593, 6.249432931750399, 6.736491630145557, 7.275729373268207, 7.677040771188120, 8.229692095606365, 8.693793154101124, 9.302342079864233, 9.530961833070760, 10.07371920024351, 10.66214422993380, 11.09615813904766, 11.59766378308502, 11.86319723384639, 12.44651132659504, 13.07994857091434

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