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Properties

Label 2-462-33.17-c1-0-7
Degree $2$
Conductor $462$
Sign $-0.925 - 0.377i$
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  + (0.309 + 0.951i)2-s + (−0.00451 + 1.73i)3-s + (−0.809 + 0.587i)4-s + (1.27 + 0.412i)5-s + (−1.64 + 0.530i)6-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s + (−2.99 − 0.0156i)9-s + 1.33i·10-s + (−1.31 + 3.04i)11-s + (−1.01 − 1.40i)12-s + (1.24 − 0.406i)13-s + (−0.587 + 0.809i)14-s + (−0.720 + 2.19i)15-s + (0.309 − 0.951i)16-s + (−1.37 + 4.21i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.00260 + 0.999i)3-s + (−0.404 + 0.293i)4-s + (0.568 + 0.184i)5-s + (−0.673 + 0.216i)6-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s + (−0.999 − 0.00521i)9-s + 0.422i·10-s + (−0.395 + 0.918i)11-s + (−0.292 − 0.405i)12-s + (0.346 − 0.112i)13-s + (−0.157 + 0.216i)14-s + (−0.186 + 0.567i)15-s + (0.0772 − 0.237i)16-s + (−0.332 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.925 - 0.377i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286104 + 1.45825i\)
\(L(\frac12)\) \(\approx\) \(0.286104 + 1.45825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.00451 - 1.73i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (1.31 - 3.04i)T \)
good5 \( 1 + (-1.27 - 0.412i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.24 + 0.406i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.37 - 4.21i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.08 + 1.48i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.58iT - 23T^{2} \)
29 \( 1 + (-4.69 + 3.41i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.679 - 2.09i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.94 + 2.86i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.50 + 5.45i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (-4.02 + 5.53i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.55 + 0.831i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.27 - 4.50i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.947 - 0.307i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + (-9.29 - 3.01i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.19 - 1.63i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.31 + 0.750i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.895 + 2.75i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.50iT - 89T^{2} \)
97 \( 1 + (-1.67 - 5.16i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.33148385870585516732559417832, −10.29630025091965493093130666268, −9.722908035099065002323220587827, −8.717821388743434724364653162294, −7.936320693168264129443241793190, −6.59592013663734557862304884674, −5.71961690570809714935028767622, −4.84818066355824414585632325606, −3.86127847117062426258335787103, −2.40728223815442759032488912302, 0.887092164814816795386351003178, 2.23153156965344030129080478265, 3.37957748899639206929155721408, 4.98984690937926828349936690833, 5.87299541343842371305022675500, 6.86954898363441411581067178519, 8.051523033931733278101363094047, 8.819066413988931204315374629656, 9.853217344420816676427225932196, 10.91617973283994731242894798091

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