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Properties

Label 2-462-21.20-c1-0-11
Degree $2$
Conductor $462$
Sign $0.936 - 0.350i$
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  + i·2-s + (0.468 − 1.66i)3-s − 4-s + 3.33·5-s + (1.66 + 0.468i)6-s + (1.56 + 2.13i)7-s − i·8-s + (−2.56 − 1.56i)9-s + 3.33i·10-s + i·11-s + (−0.468 + 1.66i)12-s + 5.20i·13-s + (−2.13 + 1.56i)14-s + (1.56 − 5.56i)15-s + 16-s + 2.39·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.270 − 0.962i)3-s − 0.5·4-s + 1.49·5-s + (0.680 + 0.191i)6-s + (0.590 + 0.807i)7-s − 0.353i·8-s + (−0.853 − 0.520i)9-s + 1.05i·10-s + 0.301i·11-s + (−0.135 + 0.481i)12-s + 1.44i·13-s + (−0.570 + 0.417i)14-s + (0.403 − 1.43i)15-s + 0.250·16-s + 0.581·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86587 + 0.337211i\)
\(L(\frac12)\) \(\approx\) \(1.86587 + 0.337211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.468 + 1.66i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
11 \( 1 - iT \)
good5 \( 1 - 3.33T + 5T^{2} \)
13 \( 1 - 5.20iT - 13T^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 + 5.73iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 4.27iT - 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 6.14T + 41T^{2} \)
43 \( 1 - 2.87T + 43T^{2} \)
47 \( 1 + 6.14T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 3.33iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 5.73T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 1.87iT - 97T^{2} \)
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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

−11.22018366381592715643542102554, −9.866747435366202056846906578029, −8.988055629908867301034725674778, −8.561987602329971155326720696792, −7.18479249556610308921494799102, −6.51288051562963522901323397220, −5.69182561151047879461790781236, −4.72346405719103189860120860320, −2.60828087285390021494490273502, −1.65627216963011825020088320753, 1.51369248127140402839567610572, 2.95729732565828514587034641264, 3.96283407624246029131825622045, 5.37902475167773046549821361403, 5.71469412198928059942309050251, 7.64340915450390320455243693622, 8.527485100095329864894775498543, 9.624495036458112079600420043186, 10.18353860901290184591208620618, 10.61645547663030296615252647219

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