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Properties

Label 2-462-21.5-c1-0-26
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 + (0.292 − 1.70i)3-s + (0.499 + 0.866i)4-s + (0.758 − 1.31i)5-s + (−1.10 + 1.33i)6-s + (−1.48 + 2.19i)7-s − 0.999i·8-s + (−2.82 − i)9-s + (−1.31 + 0.758i)10-s + (−0.866 + 0.5i)11-s + (1.62 − 0.599i)12-s − 5.44i·13-s + (2.38 − 1.15i)14-s + (−2.02 − 1.68i)15-s + (−0.5 + 0.866i)16-s + (−2.34 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.169 − 0.985i)3-s + (0.249 + 0.433i)4-s + (0.339 − 0.587i)5-s + (−0.452 + 0.543i)6-s + (−0.560 + 0.827i)7-s − 0.353i·8-s + (−0.942 − 0.333i)9-s + (−0.415 + 0.239i)10-s + (−0.261 + 0.150i)11-s + (0.469 − 0.173i)12-s − 1.51i·13-s + (0.636 − 0.308i)14-s + (−0.521 − 0.433i)15-s + (−0.125 + 0.216i)16-s + (−0.567 − 0.983i)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

\(L(1)\) \(\approx\) \(0.0557814 - 0.717128i\)
\(L(\frac12)\) \(\approx\) \(0.0557814 - 0.717128i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 1.70i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.758 + 1.31i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.44iT - 13T^{2} \)
17 \( 1 + (2.34 + 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.12 + 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.680 + 0.392i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.48iT - 29T^{2} \)
31 \( 1 + (-7.11 + 4.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.996 + 1.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 + (1.58 - 2.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.99 + 2.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.55 - 4.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.29 + 4.78i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.83 - 6.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 + (-8.74 + 5.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.25 + 2.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (-4.91 + 8.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.37iT - 97T^{2} \)
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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

−10.61294661909301729292278702707, −9.529396534215135747175180542974, −8.761023072494319063834634859778, −8.147096482875439766693481281930, −7.01756501180616665517763482977, −6.09684070712648727215527739719, −5.02810548590653839096540957308, −3.03101212568438636063746142052, −2.21794375806106394561902390486, −0.49787416708499855674186772853, 2.20106055940256670107981956303, 3.71900767486774958009604830518, 4.63150113339969085182265996726, 6.28517635214452884016185368623, 6.60517412548878677353629129612, 8.082155807053856266077982526563, 8.832353984713670870324287227488, 9.837363642722799542746258256759, 10.42317670493286668972810645410, 10.93284547778395815086023113152

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