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Properties

Label 2-462-33.29-c1-0-7
Degree $2$
Conductor $462$
Sign $0.536 - 0.843i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.856 + 1.50i)3-s + (0.309 + 0.951i)4-s + (1.72 + 2.37i)5-s + (0.191 − 1.72i)6-s + (0.951 − 0.309i)7-s + (0.309 − 0.951i)8-s + (−1.53 + 2.57i)9-s − 2.93i·10-s + (3.22 − 0.794i)11-s + (−1.16 + 1.27i)12-s + (3.69 − 5.08i)13-s + (−0.951 − 0.309i)14-s + (−2.09 + 4.62i)15-s + (−0.809 + 0.587i)16-s + (−4.68 + 3.40i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.494 + 0.869i)3-s + (0.154 + 0.475i)4-s + (0.770 + 1.06i)5-s + (0.0782 − 0.702i)6-s + (0.359 − 0.116i)7-s + (0.109 − 0.336i)8-s + (−0.510 + 0.859i)9-s − 0.927i·10-s + (0.970 − 0.239i)11-s + (−0.336 + 0.369i)12-s + (1.02 − 1.40i)13-s + (−0.254 − 0.0825i)14-s + (−0.540 + 1.19i)15-s + (−0.202 + 0.146i)16-s + (−1.13 + 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.536 - 0.843i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.536 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32004 + 0.724512i\)
\(L(\frac12)\) \(\approx\) \(1.32004 + 0.724512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.856 - 1.50i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-3.22 + 0.794i)T \)
good5 \( 1 + (-1.72 - 2.37i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-3.69 + 5.08i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.68 - 3.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.848 - 0.275i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.08iT - 23T^{2} \)
29 \( 1 + (1.40 + 4.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.16 + 3.75i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.894 - 2.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.18 - 9.80i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.94iT - 43T^{2} \)
47 \( 1 + (3.42 + 1.11i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.17 + 5.74i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-10.4 + 3.40i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.22 + 3.06i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.70T + 67T^{2} \)
71 \( 1 + (0.962 + 1.32i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.25 - 2.68i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.37 + 10.1i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.82 + 2.05i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.62iT - 89T^{2} \)
97 \( 1 + (15.5 + 11.3i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.05143145583806965812971077430, −10.23946672134726161011580610139, −9.606946312357692126074180180252, −8.633006130996853739370213242478, −7.88438756240237039991910183709, −6.54588174890128946908804522482, −5.61178111802427512807690836113, −3.96042088928612045148525053268, −3.16367873254304043110675828861, −1.88098167603641392925262474481, 1.24329179896764060628007680897, 2.09787609104692559577652873449, 4.16434073010670573612159178770, 5.43258437930874129218743435927, 6.56899496658884147201007859226, 7.07907361943284373141916228371, 8.581297908133871120604621600513, 8.936130610764783013220907677674, 9.400164142179447884444273305464, 10.96996357079601813894163684725

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