Properties

Degree $2$
Conductor $150$
Sign $1$
Motivic weight $1$
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 − 6-s + 4·7-s + 8-s + 9-s − 12-s − 2·13-s + 4·14-s + 16-s − 6·17-s + 18-s − 4·19-s − 4·21-s − 24-s − 2·26-s − 27-s + 4·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 36-s − 2·37-s − 4·38-s + 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.872·21-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.320·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49903\)
\(L(\frac12)\) \(\approx\) \(1.49903\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.01009824051402823202590914616, −11.91033651235536617108337917161, −11.23148855774643011734788423771, −10.40288135541038540368452071068, −8.749862280172923617009160642326, −7.56689807773085737112376729195, −6.38949936725272530282824625186, −5.07079211702149946545201877286, −4.30912662284802770612117339996, −2.07452981833800877942993394416, 2.07452981833800877942993394416, 4.30912662284802770612117339996, 5.07079211702149946545201877286, 6.38949936725272530282824625186, 7.56689807773085737112376729195, 8.749862280172923617009160642326, 10.40288135541038540368452071068, 11.23148855774643011734788423771, 11.91033651235536617108337917161, 13.01009824051402823202590914616

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