Properties

Label 2-462-1.1-c5-0-21
Degree $2$
Conductor $462$
Sign $-1$
Analytic cond. $74.0973$
Root an. cond. $8.60798$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s − 78·5-s + 36·6-s + 49·7-s − 64·8-s + 81·9-s + 312·10-s + 121·11-s − 144·12-s − 502·13-s − 196·14-s + 702·15-s + 256·16-s − 642·17-s − 324·18-s − 520·19-s − 1.24e3·20-s − 441·21-s − 484·22-s + 1.02e3·23-s + 576·24-s + 2.95e3·25-s + 2.00e3·26-s − 729·27-s + 784·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.39·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.986·10-s + 0.301·11-s − 0.288·12-s − 0.823·13-s − 0.267·14-s + 0.805·15-s + 1/4·16-s − 0.538·17-s − 0.235·18-s − 0.330·19-s − 0.697·20-s − 0.218·21-s − 0.213·22-s + 0.402·23-s + 0.204·24-s + 0.946·25-s + 0.582·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(74.0973\)
Root analytic conductor: \(8.60798\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{462} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 462,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
3 \( 1 + p^{2} T \)
7 \( 1 - p^{2} T \)
11 \( 1 - p^{2} T \)
good5 \( 1 + 78 T + p^{5} T^{2} \)
13 \( 1 + 502 T + p^{5} T^{2} \)
17 \( 1 + 642 T + p^{5} T^{2} \)
19 \( 1 + 520 T + p^{5} T^{2} \)
23 \( 1 - 1020 T + p^{5} T^{2} \)
29 \( 1 - 4818 T + p^{5} T^{2} \)
31 \( 1 - 1784 T + p^{5} T^{2} \)
37 \( 1 - 7958 T + p^{5} T^{2} \)
41 \( 1 - 2430 T + p^{5} T^{2} \)
43 \( 1 - 22904 T + p^{5} T^{2} \)
47 \( 1 + 11316 T + p^{5} T^{2} \)
53 \( 1 + 12222 T + p^{5} T^{2} \)
59 \( 1 + 15852 T + p^{5} T^{2} \)
61 \( 1 - 46298 T + p^{5} T^{2} \)
67 \( 1 - 19412 T + p^{5} T^{2} \)
71 \( 1 + 17292 T + p^{5} T^{2} \)
73 \( 1 + 30214 T + p^{5} T^{2} \)
79 \( 1 - 35672 T + p^{5} T^{2} \)
83 \( 1 + 43428 T + p^{5} T^{2} \)
89 \( 1 + 14934 T + p^{5} T^{2} \)
97 \( 1 - 85106 T + p^{5} T^{2} \)
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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

−9.846880211759171387293542309003, −8.794667026087743907025357977176, −7.922077008880452529700650123266, −7.23602458975580489865937036031, −6.30649004590793763338359691540, −4.87519998263780242228380733024, −4.04561550952573799096911646290, −2.58804857041186105219237659591, −0.995937273678496967266053202540, 0, 0.995937273678496967266053202540, 2.58804857041186105219237659591, 4.04561550952573799096911646290, 4.87519998263780242228380733024, 6.30649004590793763338359691540, 7.23602458975580489865937036031, 7.922077008880452529700650123266, 8.794667026087743907025357977176, 9.846880211759171387293542309003

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