Properties

Label 2-462-1.1-c3-0-30
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 − 17·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 34·10-s − 11·11-s + 12·12-s − 21·13-s + 14·14-s − 51·15-s + 16·16-s − 104·17-s + 18·18-s − 161·19-s − 68·20-s + 21·21-s − 22·22-s + 194·23-s + 24·24-s + 164·25-s − 42·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.52·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.07·10-s − 0.301·11-s + 0.288·12-s − 0.448·13-s + 0.267·14-s − 0.877·15-s + 1/4·16-s − 1.48·17-s + 0.235·18-s − 1.94·19-s − 0.760·20-s + 0.218·21-s − 0.213·22-s + 1.75·23-s + 0.204·24-s + 1.31·25-s − 0.316·26-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 \( 1 + 17 T + p^{3} T^{2} \)
13 \( 1 + 21 T + p^{3} T^{2} \)
17 \( 1 + 104 T + p^{3} T^{2} \)
19 \( 1 + 161 T + p^{3} T^{2} \)
23 \( 1 - 194 T + p^{3} T^{2} \)
29 \( 1 - 9 T + p^{3} T^{2} \)
31 \( 1 + 180 T + p^{3} T^{2} \)
37 \( 1 + 363 T + p^{3} T^{2} \)
41 \( 1 + 108 T + p^{3} T^{2} \)
43 \( 1 + 386 T + p^{3} T^{2} \)
47 \( 1 - 333 T + p^{3} T^{2} \)
53 \( 1 + 122 T + p^{3} T^{2} \)
59 \( 1 - 537 T + p^{3} T^{2} \)
61 \( 1 + 950 T + p^{3} T^{2} \)
67 \( 1 + 83 T + p^{3} T^{2} \)
71 \( 1 - 180 T + p^{3} T^{2} \)
73 \( 1 - 177 T + p^{3} T^{2} \)
79 \( 1 + 220 T + p^{3} T^{2} \)
83 \( 1 - 1112 T + p^{3} T^{2} \)
89 \( 1 + 394 T + p^{3} T^{2} \)
97 \( 1 - 826 T + p^{3} T^{2} \)
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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

−10.65364471505275467069852355884, −8.921360940819339544312945558000, −8.376268360428065645371050082194, −7.31346593574630184871125151932, −6.71102705560638171095710058270, −4.96980737227228092221512525839, −4.29530363159696233254380602318, −3.32974584145077520917040482032, −2.08688068355402639211716270103, 0, 2.08688068355402639211716270103, 3.32974584145077520917040482032, 4.29530363159696233254380602318, 4.96980737227228092221512525839, 6.71102705560638171095710058270, 7.31346593574630184871125151932, 8.376268360428065645371050082194, 8.921360940819339544312945558000, 10.65364471505275467069852355884

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