Properties

Label 2-464-1.1-c3-0-9
Degree $2$
Conductor $464$
Sign $1$
Analytic cond. $27.3768$
Root an. cond. $5.23229$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.757·3-s − 10.6·5-s + 22.1·7-s − 26.4·9-s − 39.3·11-s + 23.7·13-s − 8.07·15-s + 4.54·17-s + 155.·19-s + 16.7·21-s + 41.8·23-s − 11.4·25-s − 40.4·27-s + 29·29-s + 57.9·31-s − 29.7·33-s − 235.·35-s + 235.·37-s + 18.0·39-s − 175.·41-s + 402.·43-s + 281.·45-s − 227.·47-s + 147.·49-s + 3.44·51-s + 673.·53-s + 419.·55-s + ⋯
L(s)  = 1  + 0.145·3-s − 0.953·5-s + 1.19·7-s − 0.978·9-s − 1.07·11-s + 0.507·13-s − 0.138·15-s + 0.0648·17-s + 1.87·19-s + 0.174·21-s + 0.379·23-s − 0.0914·25-s − 0.288·27-s + 0.185·29-s + 0.335·31-s − 0.157·33-s − 1.13·35-s + 1.04·37-s + 0.0739·39-s − 0.667·41-s + 1.42·43-s + 0.932·45-s − 0.706·47-s + 0.429·49-s + 0.00944·51-s + 1.74·53-s + 1.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $1$
Analytic conductor: \(27.3768\)
Root analytic conductor: \(5.23229\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.699935043\)
\(L(\frac12)\) \(\approx\) \(1.699935043\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - 29T \)
good3 \( 1 - 0.757T + 27T^{2} \)
5 \( 1 + 10.6T + 125T^{2} \)
7 \( 1 - 22.1T + 343T^{2} \)
11 \( 1 + 39.3T + 1.33e3T^{2} \)
13 \( 1 - 23.7T + 2.19e3T^{2} \)
17 \( 1 - 4.54T + 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 - 41.8T + 1.21e4T^{2} \)
31 \( 1 - 57.9T + 2.97e4T^{2} \)
37 \( 1 - 235.T + 5.06e4T^{2} \)
41 \( 1 + 175.T + 6.89e4T^{2} \)
43 \( 1 - 402.T + 7.95e4T^{2} \)
47 \( 1 + 227.T + 1.03e5T^{2} \)
53 \( 1 - 673.T + 1.48e5T^{2} \)
59 \( 1 - 800.T + 2.05e5T^{2} \)
61 \( 1 + 222.T + 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 - 281.T + 3.57e5T^{2} \)
73 \( 1 - 1.22e3T + 3.89e5T^{2} \)
79 \( 1 + 611.T + 4.93e5T^{2} \)
83 \( 1 + 515.T + 5.71e5T^{2} \)
89 \( 1 + 358.T + 7.04e5T^{2} \)
97 \( 1 - 829.T + 9.12e5T^{2} \)
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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

−10.95583030356924640425444390547, −9.723173678106536532632278193007, −8.481707463687013210237351977198, −8.017236876011354319051527024498, −7.28051258209038325958056998714, −5.66891098212242015395114944740, −4.95550350079299034817694972694, −3.68264483548214855602328653688, −2.56075905492001820411926054467, −0.826075791967159972326976804706, 0.826075791967159972326976804706, 2.56075905492001820411926054467, 3.68264483548214855602328653688, 4.95550350079299034817694972694, 5.66891098212242015395114944740, 7.28051258209038325958056998714, 8.017236876011354319051527024498, 8.481707463687013210237351977198, 9.723173678106536532632278193007, 10.95583030356924640425444390547

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