Properties

Label 2-462-1.1-c3-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 3·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 6·10-s − 11·11-s − 12·12-s + 41·13-s + 14·14-s − 9·15-s + 16·16-s + 6·17-s + 18·18-s − 43·19-s + 12·20-s − 21·21-s − 22·22-s + 120·23-s − 24·24-s − 116·25-s + 82·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.268·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.189·10-s − 0.301·11-s − 0.288·12-s + 0.874·13-s + 0.267·14-s − 0.154·15-s + 1/4·16-s + 0.0856·17-s + 0.235·18-s − 0.519·19-s + 0.134·20-s − 0.218·21-s − 0.213·22-s + 1.08·23-s − 0.204·24-s − 0.927·25-s + 0.618·26-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: \(0\)
Selberg data: \((2,\ 462,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.897267861\)
\(L(\frac12)\) \(\approx\) \(2.897267861\)
\(L(\frac{5}{2})\) 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 \( 1 - 3 T + p^{3} T^{2} \)
13 \( 1 - 41 T + p^{3} T^{2} \)
17 \( 1 - 6 T + p^{3} T^{2} \)
19 \( 1 + 43 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 111 T + p^{3} T^{2} \)
31 \( 1 - 266 T + p^{3} T^{2} \)
37 \( 1 + 79 T + p^{3} T^{2} \)
41 \( 1 - 216 T + p^{3} T^{2} \)
43 \( 1 - 284 T + p^{3} T^{2} \)
47 \( 1 - 213 T + p^{3} T^{2} \)
53 \( 1 + 216 T + p^{3} T^{2} \)
59 \( 1 - 393 T + p^{3} T^{2} \)
61 \( 1 - 350 T + p^{3} T^{2} \)
67 \( 1 - 821 T + p^{3} T^{2} \)
71 \( 1 + 264 T + p^{3} T^{2} \)
73 \( 1 + 865 T + p^{3} T^{2} \)
79 \( 1 + 484 T + p^{3} T^{2} \)
83 \( 1 - 1158 T + p^{3} T^{2} \)
89 \( 1 - 330 T + p^{3} T^{2} \)
97 \( 1 - 980 T + p^{3} T^{2} \)
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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

−10.83179721215552060364514192548, −10.03708850732863558260252816277, −8.773928985042900908232942227260, −7.75461727740077227568890733737, −6.63034154468552784199669942755, −5.86612393159069828655749633348, −4.93023712167339422320630193086, −3.95123665416859139530933198952, −2.52411469861242553539001446013, −1.06457148104524682636197992330, 1.06457148104524682636197992330, 2.52411469861242553539001446013, 3.95123665416859139530933198952, 4.93023712167339422320630193086, 5.86612393159069828655749633348, 6.63034154468552784199669942755, 7.75461727740077227568890733737, 8.773928985042900908232942227260, 10.03708850732863558260252816277, 10.83179721215552060364514192548

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