Properties

Label 2-150-1.1-c5-0-12
Degree $2$
Conductor $150$
Sign $-1$
Analytic cond. $24.0575$
Root an. cond. $4.90485$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s − 36·6-s − 7-s − 64·8-s + 81·9-s − 210·11-s + 144·12-s − 667·13-s + 4·14-s + 256·16-s + 114·17-s − 324·18-s + 581·19-s − 9·21-s + 840·22-s − 4.35e3·23-s − 576·24-s + 2.66e3·26-s + 729·27-s − 16·28-s − 126·29-s + 7.58e3·31-s − 1.02e3·32-s − 1.89e3·33-s − 456·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.00771·7-s − 0.353·8-s + 1/3·9-s − 0.523·11-s + 0.288·12-s − 1.09·13-s + 0.00545·14-s + 1/4·16-s + 0.0956·17-s − 0.235·18-s + 0.369·19-s − 0.00445·21-s + 0.370·22-s − 1.71·23-s − 0.204·24-s + 0.774·26-s + 0.192·27-s − 0.00385·28-s − 0.0278·29-s + 1.41·31-s − 0.176·32-s − 0.302·33-s − 0.0676·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(24.0575\)
Root analytic conductor: \(4.90485\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 150,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
3 \( 1 - p^{2} T \)
5 \( 1 \)
good7 \( 1 + T + p^{5} T^{2} \)
11 \( 1 + 210 T + p^{5} T^{2} \)
13 \( 1 + 667 T + p^{5} T^{2} \)
17 \( 1 - 114 T + p^{5} T^{2} \)
19 \( 1 - 581 T + p^{5} T^{2} \)
23 \( 1 + 4350 T + p^{5} T^{2} \)
29 \( 1 + 126 T + p^{5} T^{2} \)
31 \( 1 - 7583 T + p^{5} T^{2} \)
37 \( 1 + 3742 T + p^{5} T^{2} \)
41 \( 1 + 2856 T + p^{5} T^{2} \)
43 \( 1 + 18241 T + p^{5} T^{2} \)
47 \( 1 + 23370 T + p^{5} T^{2} \)
53 \( 1 + 21684 T + p^{5} T^{2} \)
59 \( 1 + 32310 T + p^{5} T^{2} \)
61 \( 1 + 7165 T + p^{5} T^{2} \)
67 \( 1 - 59579 T + p^{5} T^{2} \)
71 \( 1 + 43080 T + p^{5} T^{2} \)
73 \( 1 + 28942 T + p^{5} T^{2} \)
79 \( 1 - 27608 T + p^{5} T^{2} \)
83 \( 1 + 1782 T + p^{5} T^{2} \)
89 \( 1 - 50208 T + p^{5} T^{2} \)
97 \( 1 - 142793 T + p^{5} T^{2} \)
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

−11.58367188932704811794021752768, −10.16223086079876499335452588070, −9.700501931417987180744364090900, −8.336720911792825821994889627327, −7.67445350059577524699688081466, −6.42637810910469364890500434719, −4.84634474047403086297118170014, −3.13665459306655040049505795703, −1.86229455487229835818126326278, 0, 1.86229455487229835818126326278, 3.13665459306655040049505795703, 4.84634474047403086297118170014, 6.42637810910469364890500434719, 7.67445350059577524699688081466, 8.336720911792825821994889627327, 9.700501931417987180744364090900, 10.16223086079876499335452588070, 11.58367188932704811794021752768

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