Properties

Label 2-150-1.1-c3-0-5
Degree $2$
Conductor $150$
Sign $1$
Analytic cond. $8.85028$
Root an. cond. $2.97494$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s + 6·6-s + 16·7-s + 8·8-s + 9·9-s + 12·11-s + 12·12-s − 38·13-s + 32·14-s + 16·16-s + 126·17-s + 18·18-s + 20·19-s + 48·21-s + 24·22-s − 168·23-s + 24·24-s − 76·26-s + 27·27-s + 64·28-s + 30·29-s − 88·31-s + 32·32-s + 36·33-s + 252·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.863·7-s + 0.353·8-s + 1/3·9-s + 0.328·11-s + 0.288·12-s − 0.810·13-s + 0.610·14-s + 1/4·16-s + 1.79·17-s + 0.235·18-s + 0.241·19-s + 0.498·21-s + 0.232·22-s − 1.52·23-s + 0.204·24-s − 0.573·26-s + 0.192·27-s + 0.431·28-s + 0.192·29-s − 0.509·31-s + 0.176·32-s + 0.189·33-s + 1.27·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.282471807\)
\(L(\frac12)\) \(\approx\) \(3.282471807\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
5 \( 1 \)
good7 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 - 126 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 - 30 T + p^{3} T^{2} \)
31 \( 1 + 88 T + p^{3} T^{2} \)
37 \( 1 + 254 T + p^{3} T^{2} \)
41 \( 1 - 42 T + p^{3} T^{2} \)
43 \( 1 - 52 T + p^{3} T^{2} \)
47 \( 1 - 96 T + p^{3} T^{2} \)
53 \( 1 + 198 T + p^{3} T^{2} \)
59 \( 1 + 660 T + p^{3} T^{2} \)
61 \( 1 + 538 T + p^{3} T^{2} \)
67 \( 1 + 884 T + p^{3} T^{2} \)
71 \( 1 - 792 T + p^{3} T^{2} \)
73 \( 1 + 218 T + p^{3} T^{2} \)
79 \( 1 + 520 T + p^{3} T^{2} \)
83 \( 1 - 492 T + p^{3} T^{2} \)
89 \( 1 - 810 T + p^{3} T^{2} \)
97 \( 1 + 1154 T + p^{3} T^{2} \)
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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

−12.40884971972922527309970470616, −11.91314201878579295207161256564, −10.56213790076189419667827175894, −9.552100024663980302549636636753, −8.103350576543129974640002257728, −7.37493600032744107877365840821, −5.80046633911998718669886850555, −4.62381030957146324738871678445, −3.31052713369813163270276767745, −1.73220558244963274682327977520, 1.73220558244963274682327977520, 3.31052713369813163270276767745, 4.62381030957146324738871678445, 5.80046633911998718669886850555, 7.37493600032744107877365840821, 8.103350576543129974640002257728, 9.552100024663980302549636636753, 10.56213790076189419667827175894, 11.91314201878579295207161256564, 12.40884971972922527309970470616

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