Properties

Label 2-462-77.6-c1-0-14
Degree $2$
Conductor $462$
Sign $-0.451 + 0.892i$
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  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−2.13 − 0.693i)5-s + (0.309 − 0.951i)6-s + (−2.54 − 0.709i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − 2.24·10-s + (1.68 − 2.85i)11-s − 0.999i·12-s + (−0.720 − 2.21i)13-s + (−2.64 + 0.112i)14-s + (−1.81 + 1.31i)15-s + (0.309 − 0.951i)16-s + (0.988 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.953 − 0.309i)5-s + (0.126 − 0.388i)6-s + (−0.963 − 0.268i)7-s + (0.207 − 0.286i)8-s + (−0.103 − 0.317i)9-s − 0.709·10-s + (0.508 − 0.861i)11-s − 0.288i·12-s + (−0.199 − 0.614i)13-s + (−0.706 + 0.0300i)14-s + (−0.468 + 0.340i)15-s + (0.0772 − 0.237i)16-s + (0.239 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868296 - 1.41288i\)
\(L(\frac12)\) \(\approx\) \(0.868296 - 1.41288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (2.54 + 0.709i)T \)
11 \( 1 + (-1.68 + 2.85i)T \)
good5 \( 1 + (2.13 + 0.693i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.720 + 2.21i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.988 + 3.04i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.0237 - 0.0172i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + (-4.01 - 5.52i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.88 - 0.612i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.68 - 1.94i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.83 - 6.42i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.48iT - 43T^{2} \)
47 \( 1 + (-0.343 + 0.473i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.39 + 7.37i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.23 - 7.21i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.44 + 13.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.48T + 67T^{2} \)
71 \( 1 + (-3.72 + 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.0950 - 0.0690i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (15.6 - 5.07i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.28 + 7.02i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (-12.3 + 4.00i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.05234332259933085352534348918, −9.902672003787383953218590545162, −8.907066080277001097824680299870, −7.908842183277415865872046186676, −7.00764661582959753070763713125, −6.12477121634097562728517469691, −4.84880825774482919047333702270, −3.58960357592157069850382588485, −2.95659715279402307801038014337, −0.813268816975542724987998154113, 2.46172402371390215037196670774, 3.73395496355552736164466935109, 4.24549753070767102576236560461, 5.64529653214626086173623990259, 6.79987627732185100244274953853, 7.45654793479963653838597232881, 8.640968026775700748974504368861, 9.548989346306007953644079822510, 10.47783639457689406711122895240, 11.52778425418067018894695452328

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