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Properties

Label 2-462-77.41-c1-0-12
Degree $2$
Conductor $462$
Sign $0.952 - 0.303i$
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  + (0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (1.30 − 1.79i)5-s + (0.809 + 0.587i)6-s + (2.63 + 0.236i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + 2.21·10-s + (−3.31 + 0.0448i)11-s + i·12-s + (1.39 − 1.01i)13-s + (1.35 + 2.27i)14-s + (0.684 − 2.10i)15-s + (−0.809 − 0.587i)16-s + (2.79 + 2.03i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.582 − 0.801i)5-s + (0.330 + 0.239i)6-s + (0.996 + 0.0892i)7-s + (−0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + 0.700·10-s + (−0.999 + 0.0135i)11-s + 0.288i·12-s + (0.386 − 0.280i)13-s + (0.362 + 0.606i)14-s + (0.176 − 0.544i)15-s + (−0.202 − 0.146i)16-s + (0.679 + 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30692 + 0.357979i\)
\(L(\frac12)\) \(\approx\) \(2.30692 + 0.357979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (-2.63 - 0.236i)T \)
11 \( 1 + (3.31 - 0.0448i)T \)
good5 \( 1 + (-1.30 + 1.79i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.39 + 1.01i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.79 - 2.03i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.22 + 3.75i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.14T + 23T^{2} \)
29 \( 1 + (1.17 + 0.381i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.89 - 3.98i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.77 - 8.54i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.153 - 0.471i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.724iT - 43T^{2} \)
47 \( 1 + (5.80 - 1.88i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.34 - 3.15i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (13.4 + 4.38i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.61 + 6.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + (0.667 + 0.485i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.733 - 2.25i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.00 + 1.37i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.44 + 4.68i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.56iT - 89T^{2} \)
97 \( 1 + (-9.02 - 12.4i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.13007950638929696608126451099, −10.09290355128475576684900454522, −8.987317137006495878973351924419, −8.241778283693797965640547820797, −7.67885682096143791638626158685, −6.33063856240691395914123607825, −5.26057697200227462325223065103, −4.63665271761012164809150764732, −3.08553661899945834273259955969, −1.61279296952521545999844482098, 1.82688624809862759295757584947, 2.80571295991311199549352060135, 4.01263765167109910789846424071, 5.17292991383822753697977965059, 6.12704962537022861067037916623, 7.48892358918744726722758075133, 8.266021709851110464400376316430, 9.462889649893664909892600892671, 10.34257147324810416144674889501, 10.82051672807923887777798089011

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