Properties

Label 2-327990-1.1-c1-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 14-s − 15-s + 16-s + 2·17-s − 18-s − 6·19-s + 20-s − 21-s − 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 + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-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 − 0.218·21-s − 0.213·22-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: $\chi_{327990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.062450083\)
\(L(\frac12)\) \(\approx\) \(1.062450083\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.44002646327941, −12.20793992076939, −11.52527361494900, −11.20855357068844, −10.73723040396427, −10.35765565510593, −9.928770609321851, −9.354117120088970, −9.110223791310357, −8.431033189498246, −8.003977187845903, −7.659882299206767, −6.890981706316851, −6.591986769160194, −6.207355012873463, −5.630109678283355, −5.023871013964003, −4.755458601766229, −3.961681972724128, −3.499184525941312, −2.729780661400160, −2.136725167038443, −1.667287332784709, −1.090091736451467, −0.3369793495679646, 0.3369793495679646, 1.090091736451467, 1.667287332784709, 2.136725167038443, 2.729780661400160, 3.499184525941312, 3.961681972724128, 4.755458601766229, 5.023871013964003, 5.630109678283355, 6.207355012873463, 6.591986769160194, 6.890981706316851, 7.659882299206767, 8.003977187845903, 8.431033189498246, 9.110223791310357, 9.354117120088970, 9.928770609321851, 10.35765565510593, 10.73723040396427, 11.20855357068844, 11.52527361494900, 12.20793992076939, 12.44002646327941

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