Properties

Label 2-327990-1.1-c1-0-31
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 + 2·7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 2·17-s − 18-s + 6·19-s − 20-s + 2·21-s + 2·22-s − 2·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-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 − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s − 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-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: $\chi_{327990} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−12.77117732672810, −12.27084460926433, −11.84232136303131, −11.43961452855406, −10.97815102709077, −10.52972058969347, −10.00759617375798, −9.688550600194321, −9.076771053470867, −8.658042553315397, −8.220934557029555, −7.821262730713152, −7.409864533360804, −7.141628114209042, −6.417811349994533, −5.746640771484761, −5.344501145843146, −4.821358790807255, −4.194038022279773, −3.581557168977493, −3.164377855857212, −2.585867568983225, −1.967228137462458, −1.374064749234271, −0.8592595433475485, 0, 0.8592595433475485, 1.374064749234271, 1.967228137462458, 2.585867568983225, 3.164377855857212, 3.581557168977493, 4.194038022279773, 4.821358790807255, 5.344501145843146, 5.746640771484761, 6.417811349994533, 7.141628114209042, 7.409864533360804, 7.821262730713152, 8.220934557029555, 8.658042553315397, 9.076771053470867, 9.688550600194321, 10.00759617375798, 10.52972058969347, 10.97815102709077, 11.43961452855406, 11.84232136303131, 12.27084460926433, 12.77117732672810

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