Properties

Label 2-462-1.1-c3-0-16
Degree $2$
Conductor $462$
Sign $-1$
Analytic cond. $27.2588$
Root an. cond. $5.22100$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 14·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 28·10-s + 11·11-s − 12·12-s + 38·13-s + 14·14-s + 42·15-s + 16·16-s + 54·17-s − 18·18-s + 40·19-s − 56·20-s + 21·21-s − 22·22-s + 8·23-s + 24·24-s + 71·25-s − 76·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.25·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.885·10-s + 0.301·11-s − 0.288·12-s + 0.810·13-s + 0.267·14-s + 0.722·15-s + 1/4·16-s + 0.770·17-s − 0.235·18-s + 0.482·19-s − 0.626·20-s + 0.218·21-s − 0.213·22-s + 0.0725·23-s + 0.204·24-s + 0.567·25-s − 0.573·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(27.2588\)
Root analytic conductor: \(5.22100\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{462} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 462,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 + p T \)
11 \( 1 - p T \)
good5 \( 1 + 14 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 - 8 T + p^{3} T^{2} \)
29 \( 1 + 170 T + p^{3} T^{2} \)
31 \( 1 - 92 T + p^{3} T^{2} \)
37 \( 1 - 294 T + p^{3} T^{2} \)
41 \( 1 + 258 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 + 76 T + p^{3} T^{2} \)
53 \( 1 + 322 T + p^{3} T^{2} \)
59 \( 1 - 260 T + p^{3} T^{2} \)
61 \( 1 - 22 T + p^{3} T^{2} \)
67 \( 1 + 436 T + p^{3} T^{2} \)
71 \( 1 + 368 T + p^{3} T^{2} \)
73 \( 1 + 2 T + p^{3} T^{2} \)
79 \( 1 + 200 T + p^{3} T^{2} \)
83 \( 1 + 952 T + p^{3} T^{2} \)
89 \( 1 + 70 T + p^{3} T^{2} \)
97 \( 1 + 1086 T + p^{3} T^{2} \)
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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

−10.20981724107120844238445757808, −9.328686112958773256832869935282, −8.255829834660828174237478345427, −7.54894234437512205701807755914, −6.62243479502536798864921205430, −5.60271201056860779821802176162, −4.16547927711364497833153259771, −3.20441939737899223097427219198, −1.23068841121845597614546538997, 0, 1.23068841121845597614546538997, 3.20441939737899223097427219198, 4.16547927711364497833153259771, 5.60271201056860779821802176162, 6.62243479502536798864921205430, 7.54894234437512205701807755914, 8.255829834660828174237478345427, 9.328686112958773256832869935282, 10.20981724107120844238445757808

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