Properties

Label 2-462-21.5-c1-0-25
Degree $2$
Conductor $462$
Sign $0.987 - 0.154i$
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 + (1.70 + 0.292i)3-s + (0.499 + 0.866i)4-s + (1.46 − 2.53i)5-s + (1.33 + 1.10i)6-s + (−2.19 − 1.48i)7-s + 0.999i·8-s + (2.82 + i)9-s + (2.53 − 1.46i)10-s + (0.866 − 0.5i)11-s + (0.599 + 1.62i)12-s + 5.44i·13-s + (−1.15 − 2.38i)14-s + (3.24 − 3.90i)15-s + (−0.5 + 0.866i)16-s + (−2.65 − 4.60i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.985 + 0.169i)3-s + (0.249 + 0.433i)4-s + (0.655 − 1.13i)5-s + (0.543 + 0.452i)6-s + (−0.827 − 0.560i)7-s + 0.353i·8-s + (0.942 + 0.333i)9-s + (0.802 − 0.463i)10-s + (0.261 − 0.150i)11-s + (0.173 + 0.469i)12-s + 1.51i·13-s + (−0.308 − 0.636i)14-s + (0.838 − 1.00i)15-s + (−0.125 + 0.216i)16-s + (−0.644 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.987 - 0.154i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.987 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74897 + 0.213827i\)
\(L(\frac12)\) \(\approx\) \(2.74897 + 0.213827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.292i)T \)
7 \( 1 + (2.19 + 1.48i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-1.46 + 2.53i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.44iT - 13T^{2} \)
17 \( 1 + (2.65 + 4.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.544 + 0.314i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.90 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.31iT - 29T^{2} \)
31 \( 1 + (6.56 - 3.79i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.22 - 3.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 + 4.14T + 43T^{2} \)
47 \( 1 + (4.41 - 7.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.33 + 3.65i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.117 - 0.203i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.60 + 3.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.31iT - 71T^{2} \)
73 \( 1 + (-11.8 + 6.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.47 - 7.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.02T + 83T^{2} \)
89 \( 1 + (-0.987 + 1.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.2iT - 97T^{2} \)
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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

−11.16785251985700899873419835203, −9.724020551587106400716570721774, −9.263444535195137679949492898178, −8.559033864571735297803010155037, −7.19458569652678004166224552153, −6.58760329328608552146467100226, −5.09826931974353368157465419776, −4.33281720278369883744600116522, −3.21938200662558958455642796019, −1.73592423675866534715438604018, 2.04021668865249195926769188036, 2.98062704818337645250234033732, 3.67363274816846404908980807067, 5.40078512482620023293116008026, 6.45200213664880141190076844514, 7.06991370822887716212937541038, 8.441660490863421573914142303197, 9.345316142466276580236914434910, 10.36649593366501016517274146647, 10.67468995319188514878698030906

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