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Properties

Label 2-462-33.32-c1-0-19
Degree $2$
Conductor $462$
Sign $0.978 + 0.207i$
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  + 2-s + (1.37 + 1.05i)3-s + 4-s − 3.92i·5-s + (1.37 + 1.05i)6-s + i·7-s + 8-s + (0.775 + 2.89i)9-s − 3.92i·10-s + (−1.42 − 2.99i)11-s + (1.37 + 1.05i)12-s − 0.109i·13-s + i·14-s + (4.14 − 5.39i)15-s + 16-s + 6.97·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.793 + 0.608i)3-s + 0.5·4-s − 1.75i·5-s + (0.560 + 0.430i)6-s + 0.377i·7-s + 0.353·8-s + (0.258 + 0.965i)9-s − 1.24i·10-s + (−0.430 − 0.902i)11-s + (0.396 + 0.304i)12-s − 0.0302i·13-s + 0.267i·14-s + (1.06 − 1.39i)15-s + 0.250·16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68162 - 0.281676i\)
\(L(\frac12)\) \(\approx\) \(2.68162 - 0.281676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.37 - 1.05i)T \)
7 \( 1 - iT \)
11 \( 1 + (1.42 + 2.99i)T \)
good5 \( 1 + 3.92iT - 5T^{2} \)
13 \( 1 + 0.109iT - 13T^{2} \)
17 \( 1 - 6.97T + 17T^{2} \)
19 \( 1 - 5.78iT - 19T^{2} \)
23 \( 1 + 0.938iT - 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 - 7.21T + 37T^{2} \)
41 \( 1 + 8.95T + 41T^{2} \)
43 \( 1 - 1.44iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 5.55iT - 53T^{2} \)
59 \( 1 - 7.60iT - 59T^{2} \)
61 \( 1 + 6.35iT - 61T^{2} \)
67 \( 1 - 5.60T + 67T^{2} \)
71 \( 1 + 4.39iT - 71T^{2} \)
73 \( 1 - 3.31iT - 73T^{2} \)
79 \( 1 + 0.361iT - 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + 0.722iT - 89T^{2} \)
97 \( 1 + 3.48T + 97T^{2} \)
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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

−11.11889147242564190651454666779, −9.918981612949747025449075973948, −9.250260426497602149549453640197, −8.165981692227781631375777994506, −7.84949984276451969285555268439, −5.68887698141195406266729241251, −5.35513832329294414578413211118, −4.15922704546247809363293586548, −3.26039977561571755555978404721, −1.60908759030253174622253861528, 2.06548055137123558921920072923, 3.05870302742654619027027830745, 3.84004568112856240430263248887, 5.50217011549406991390891352990, 6.74103307118721843846100019470, 7.26144883462653383752410721469, 7.85421886962103693607627348888, 9.515742730473287814106010832036, 10.23443225459578875963507986711, 11.18239166078312965133208599926

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