Properties

Label 2-462-77.9-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.989 - 0.147i$
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  + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−0.408 − 0.0868i)5-s + (−0.309 − 0.951i)6-s + (−2.24 − 1.40i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.208 − 0.361i)10-s + (−3.16 + 0.992i)11-s + (0.5 − 0.866i)12-s + (−1.95 + 6.00i)13-s + (−0.453 − 2.60i)14-s + (0.338 + 0.245i)15-s + (−0.978 − 0.207i)16-s + (−4.10 + 4.55i)17-s + ⋯
L(s)  = 1  + (0.473 + 0.525i)2-s + (−0.527 − 0.234i)3-s + (−0.0522 + 0.497i)4-s + (−0.182 − 0.0388i)5-s + (−0.126 − 0.388i)6-s + (−0.846 − 0.531i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (−0.0660 − 0.114i)10-s + (−0.954 + 0.299i)11-s + (0.144 − 0.249i)12-s + (−0.541 + 1.66i)13-s + (−0.121 − 0.696i)14-s + (0.0872 + 0.0634i)15-s + (−0.244 − 0.0519i)16-s + (−0.994 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0360441 + 0.487403i\)
\(L(\frac12)\) \(\approx\) \(0.0360441 + 0.487403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (2.24 + 1.40i)T \)
11 \( 1 + (3.16 - 0.992i)T \)
good5 \( 1 + (0.408 + 0.0868i)T + (4.56 + 2.03i)T^{2} \)
13 \( 1 + (1.95 - 6.00i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.10 - 4.55i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (-0.354 - 3.37i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-4.39 + 7.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.21 + 2.33i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.03 + 0.433i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-0.784 + 0.349i)T + (24.7 - 27.4i)T^{2} \)
41 \( 1 + (3.13 - 2.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.07T + 43T^{2} \)
47 \( 1 + (-0.0192 - 0.182i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (6.25 - 1.33i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (0.467 - 4.45i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-6.67 - 1.41i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-5.77 - 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.960 + 2.95i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.416 + 3.95i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-8.97 - 9.96i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (3.63 + 11.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.82 + 8.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.35 + 7.26i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.60783257193678698468280194748, −10.63326357089261060854929993957, −9.773405705193766897847149309373, −8.613700694539763861743019788608, −7.54737242236255877127995072285, −6.70219650560250478625973941607, −6.09760757098954995398482788132, −4.70029684287891918431285623881, −3.99689845169061809339392615555, −2.29883784465159286416998647688, 0.25577394020608770423536392284, 2.64278112880713296550004596333, 3.44931986586168465262870246286, 5.14544864117536332524590469869, 5.42863991636034443013103762141, 6.75723284904646470249215873532, 7.78308564272274998300856488108, 9.212477667067343923199642233511, 9.818824462343332420483571407505, 10.84913887548199696803840883138

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