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Properties

Label 2-462-231.32-c1-0-26
Degree $2$
Conductor $462$
Sign $-0.113 + 0.993i$
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  + (0.5 + 0.866i)2-s + (−1.20 + 1.24i)3-s + (−0.499 + 0.866i)4-s + (−1.00 + 0.580i)5-s + (−1.67 − 0.426i)6-s + (−2.19 − 1.47i)7-s − 0.999·8-s + (−0.0797 − 2.99i)9-s + (−1.00 − 0.580i)10-s + (−2.34 + 2.34i)11-s + (−0.470 − 1.66i)12-s − 3.56i·13-s + (0.173 − 2.64i)14-s + (0.494 − 1.94i)15-s + (−0.5 − 0.866i)16-s + (1.71 − 2.96i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.697 + 0.716i)3-s + (−0.249 + 0.433i)4-s + (−0.449 + 0.259i)5-s + (−0.685 − 0.173i)6-s + (−0.831 − 0.555i)7-s − 0.353·8-s + (−0.0265 − 0.999i)9-s + (−0.318 − 0.183i)10-s + (−0.705 + 0.708i)11-s + (−0.135 − 0.481i)12-s − 0.989i·13-s + (0.0463 − 0.705i)14-s + (0.127 − 0.503i)15-s + (−0.125 − 0.216i)16-s + (0.415 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.113 + 0.993i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0306713 - 0.0343749i\)
\(L(\frac12)\) \(\approx\) \(0.0306713 - 0.0343749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.20 - 1.24i)T \)
7 \( 1 + (2.19 + 1.47i)T \)
11 \( 1 + (2.34 - 2.34i)T \)
good5 \( 1 + (1.00 - 0.580i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.56iT - 13T^{2} \)
17 \( 1 + (-1.71 + 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.30 + 2.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.67 - 4.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 + (0.547 - 0.947i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.31 + 9.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 + 6.97iT - 43T^{2} \)
47 \( 1 + (4.23 - 2.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.75 - 1.01i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.87 + 5.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.96 - 5.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.911 - 1.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (8.95 + 5.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.06 - 4.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.21T + 83T^{2} \)
89 \( 1 + (2.99 - 1.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−10.74353265148611206821145413570, −9.913156885691658195246599003292, −9.259699733958546064972872508159, −7.54144881621206113138236576980, −7.30840097565312760801917459091, −5.85743939375951888038188572004, −5.25721298248428753966833231720, −3.97973889611588349956244723194, −3.19837411841949412285902908278, −0.02650352585896147144231977855, 1.83907669989090722783495017593, 3.24671002832283068259372907104, 4.53664839718268032059477493939, 5.81172076421333847886217643095, 6.26578192116009160636227741595, 7.68429985108290895109567026312, 8.510732199065595941776682338421, 9.764889952356554521241937479952, 10.54624459572055758964582926691, 11.64327367445615590012391905527

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