Properties

Label 2-462-21.20-c1-0-8
Degree $2$
Conductor $462$
Sign $-0.487 - 0.872i$
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  + i·2-s + (0.618 + 1.61i)3-s − 4-s + 1.23·5-s + (−1.61 + 0.618i)6-s + (2.61 − 0.381i)7-s i·8-s + (−2.23 + 2.00i)9-s + 1.23i·10-s i·11-s + (−0.618 − 1.61i)12-s + 6i·13-s + (0.381 + 2.61i)14-s + (0.763 + 2.00i)15-s + 16-s + 2.76·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.356 + 0.934i)3-s − 0.5·4-s + 0.552·5-s + (−0.660 + 0.252i)6-s + (0.989 − 0.144i)7-s − 0.353i·8-s + (−0.745 + 0.666i)9-s + 0.390i·10-s − 0.301i·11-s + (−0.178 − 0.467i)12-s + 1.66i·13-s + (0.102 + 0.699i)14-s + (0.197 + 0.516i)15-s + 0.250·16-s + 0.670·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.487 - 0.872i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871863 + 1.48623i\)
\(L(\frac12)\) \(\approx\) \(0.871863 + 1.48623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.618 - 1.61i)T \)
7 \( 1 + (-2.61 + 0.381i)T \)
11 \( 1 + iT \)
good5 \( 1 - 1.23T + 5T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 + 8.47iT - 29T^{2} \)
31 \( 1 - 5.23iT - 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 5.70iT - 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 + 8.18iT - 73T^{2} \)
79 \( 1 - 2.76T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 4.94T + 89T^{2} \)
97 \( 1 - 6.47iT - 97T^{2} \)
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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

−11.26193858753794294011789602936, −10.17403937152982549605093155466, −9.531146501281118413878703356241, −8.571277210422102481087504670426, −7.946442717532163595655263088628, −6.64786886128596789253957281411, −5.58059461009239174623110925924, −4.67859075981237580197284009778, −3.79730004063956671084240800959, −2.02024850973126980547445189196, 1.17292470199172224556447731944, 2.31279278911285762268133530575, 3.43125204184876553210421189297, 5.12833905337014648757307953939, 5.85051485121645107465174064908, 7.36203227983659590319131766139, 8.043201032867115936847951399771, 8.917263915757782937350626342501, 9.938996810374542909397110885630, 10.82427005422884450355433478564

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