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Properties

Label 2-462-21.17-c1-0-27
Degree $2$
Conductor $462$
Sign $-0.861 - 0.507i$
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  + (0.866 − 0.5i)2-s + (−1.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−2.09 − 3.62i)5-s − 1.73·6-s + (−1.62 + 2.09i)7-s − 0.999i·8-s + (1.5 + 2.59i)9-s + (−3.62 − 2.09i)10-s + (0.866 + 0.5i)11-s + (−1.49 + 0.866i)12-s − 3.46i·13-s + (−0.358 + 2.62i)14-s + 7.24i·15-s + (−0.5 − 0.866i)16-s + (−1.58 + 2.74i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.866 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.935 − 1.61i)5-s − 0.707·6-s + (−0.612 + 0.790i)7-s − 0.353i·8-s + (0.5 + 0.866i)9-s + (−1.14 − 0.661i)10-s + (0.261 + 0.150i)11-s + (−0.433 + 0.250i)12-s − 0.960i·13-s + (−0.0958 + 0.700i)14-s + 1.87i·15-s + (−0.125 − 0.216i)16-s + (−0.384 + 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.861 - 0.507i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140771 + 0.515809i\)
\(L(\frac12)\) \(\approx\) \(0.140771 + 0.515809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (1.62 - 2.09i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (1.58 - 2.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.24 - 4.18i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 0.621i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + (-1.24 - 0.717i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + (2.09 + 3.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.34 + 4.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.16 + 5.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.621 + 0.358i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.74 + 3.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.62 - 4.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (-1.01 - 1.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.297iT - 97T^{2} \)
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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

−10.83224323998880306477593803461, −9.766841462558956215832290631105, −8.543381849659394583400186064439, −7.919984243742005656681855015036, −6.42144813839286324658251719769, −5.68436398649787769280703491309, −4.73757996558183855403851288585, −3.84638528839714958078243365836, −1.92014146848570437366186590542, −0.29034596771996391431861925852, 2.95319371466308379878130323782, 3.94389553175360471993935148527, 4.62979179186096699341518215274, 6.36500885038805036966907603648, 6.73871562962147064221267580385, 7.34385733045572950710919239870, 8.917120435441861089502842404514, 10.15475725599030664149006824778, 11.01346563563536076402089747251, 11.28240100794054857399270139477

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