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Properties

Label 2-462-33.29-c1-0-12
Degree $2$
Conductor $462$
Sign $-0.295 + 0.955i$
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  + (−0.809 − 0.587i)2-s + (−1.67 − 0.427i)3-s + (0.309 + 0.951i)4-s + (0.354 + 0.487i)5-s + (1.10 + 1.33i)6-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (2.63 + 1.43i)9-s − 0.602i·10-s + (−3.17 − 0.964i)11-s + (−0.112 − 1.72i)12-s + (0.688 − 0.947i)13-s + (0.951 + 0.309i)14-s + (−0.386 − 0.969i)15-s + (−0.809 + 0.587i)16-s + (4.38 − 3.18i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.969 − 0.246i)3-s + (0.154 + 0.475i)4-s + (0.158 + 0.217i)5-s + (0.451 + 0.543i)6-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + (0.878 + 0.478i)9-s − 0.190i·10-s + (−0.956 − 0.290i)11-s + (−0.0324 − 0.498i)12-s + (0.190 − 0.262i)13-s + (0.254 + 0.0825i)14-s + (−0.0997 − 0.250i)15-s + (−0.202 + 0.146i)16-s + (1.06 − 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.346193 - 0.469328i\)
\(L(\frac12)\) \(\approx\) \(0.346193 - 0.469328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (1.67 + 0.427i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (3.17 + 0.964i)T \)
good5 \( 1 + (-0.354 - 0.487i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-0.688 + 0.947i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.38 + 3.18i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-3.55 - 1.15i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.43iT - 23T^{2} \)
29 \( 1 + (2.42 + 7.46i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.43 + 6.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.74 + 11.5i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.34 - 7.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.23iT - 43T^{2} \)
47 \( 1 + (-9.20 - 2.99i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.24 + 7.21i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.10 - 1.98i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.20 - 5.78i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 + (2.83 + 3.90i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.39 + 0.454i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.11 + 8.42i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.7 + 7.80i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + (4.11 + 2.98i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.73917917910350685693766589944, −10.06088507567242944058240671941, −9.278422224780141950746726337737, −7.79955383419024221642398801440, −7.35094108294305037017426112747, −5.96623214101941125494371687758, −5.33236135686696491575195297589, −3.70286901489611987120792391587, −2.31267562124440520012705694255, −0.53295447152239921276747606382, 1.36628524399334873250330181015, 3.46930987941198426614046178303, 5.08048491982251554789233018423, 5.56404231715296586878516761159, 6.79409389310918884246479531844, 7.46304439418803904397529908474, 8.723663875606928197752147087987, 9.627184338818015627709845473312, 10.45068106861055697207737980234, 10.95749843069007842154914399379

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