Properties

Label 2-462-11.4-c1-0-7
Degree $2$
Conductor $462$
Sign $-0.836 + 0.548i$
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 + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−2.65 + 1.92i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 3.28·10-s + (2.53 + 2.13i)11-s − 12-s + (−2.52 − 1.83i)13-s + (−0.309 + 0.951i)14-s + (−1.01 − 3.11i)15-s + (−0.809 + 0.587i)16-s + (−3.18 + 2.31i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−1.18 + 0.862i)5-s + (0.330 − 0.239i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + 1.03·10-s + (0.764 + 0.644i)11-s − 0.288·12-s + (−0.699 − 0.508i)13-s + (−0.0825 + 0.254i)14-s + (−0.261 − 0.805i)15-s + (−0.202 + 0.146i)16-s + (−0.773 + 0.562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.836 + 0.548i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0213768 - 0.0716251i\)
\(L(\frac12)\) \(\approx\) \(0.0213768 - 0.0716251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.53 - 2.13i)T \)
good5 \( 1 + (2.65 - 1.92i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.52 + 1.83i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.18 - 2.31i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.74 + 5.36i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.33T + 23T^{2} \)
29 \( 1 + (3.11 + 9.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (8.56 + 6.22i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.249 - 0.768i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.864 + 2.66i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.17T + 43T^{2} \)
47 \( 1 + (3.52 - 10.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.41 - 1.02i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.68 + 8.26i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.12 - 5.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (2.85 - 2.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.196 - 0.603i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.41 - 6.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.5 - 9.08i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.99T + 89T^{2} \)
97 \( 1 + (-11.0 - 8.02i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.85663840921287924779442700458, −9.816371941044134259730682922688, −9.158926365628111567273358016072, −7.77940342396964083196767827866, −7.32979453111625836187151385672, −6.18760227713615111442494000619, −4.40224968919230912356953890683, −3.79824358600560736851313268079, −2.48891271095644387884175389610, −0.05660711863752189011835790835, 1.60447331561164306161180348848, 3.59042764927761029864947039138, 4.86799227194430527831477217195, 5.92121970097444575966143649939, 7.08289684019498216557720644112, 7.72197404480043650722299113349, 8.809502566638497680558093313657, 9.115588827917899703764160500085, 10.58090330045012342895167258527, 11.64489441541460163168464821999

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