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Properties

Label 2-462-77.6-c1-0-15
Degree $2$
Conductor $462$
Sign $-0.946 + 0.323i$
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 + (−2.46 − 0.800i)5-s + (−0.309 + 0.951i)6-s + (−2.50 − 0.857i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − 2.59·10-s + (−3.14 + 1.05i)11-s + 0.999i·12-s + (−0.400 − 1.23i)13-s + (−2.64 − 0.0416i)14-s + (2.09 − 1.52i)15-s + (0.309 − 0.951i)16-s + (−0.168 + 0.518i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (−1.10 − 0.357i)5-s + (−0.126 + 0.388i)6-s + (−0.946 − 0.323i)7-s + (0.207 − 0.286i)8-s + (−0.103 − 0.317i)9-s − 0.819·10-s + (−0.948 + 0.317i)11-s + 0.288i·12-s + (−0.111 − 0.341i)13-s + (−0.707 − 0.0111i)14-s + (0.541 − 0.393i)15-s + (0.0772 − 0.237i)16-s + (−0.0408 + 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.946 + 0.323i$
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: \((2,\ 462,\ (\ :1/2),\ -0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0625236 - 0.376284i\)
\(L(\frac12)\) \(\approx\) \(0.0625236 - 0.376284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.857i)T \)
11 \( 1 + (3.14 - 1.05i)T \)
good5 \( 1 + (2.46 + 0.800i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.400 + 1.23i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.168 - 0.518i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.23 + 3.07i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 + (-3.42 - 4.71i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.50 + 1.78i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.42 + 2.49i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.60 + 4.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.99iT - 43T^{2} \)
47 \( 1 + (-0.750 + 1.03i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.67 - 5.14i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.190 - 0.262i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.10 - 9.54i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + (2.97 - 9.15i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.93 + 5.04i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.41 + 1.43i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.74 - 5.37i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 15.6iT - 89T^{2} \)
97 \( 1 + (-8.14 + 2.64i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.53635595010189586075206433099, −10.21845352415267387731872931930, −8.873407024954452645497421216339, −7.81428820837829579953944556807, −6.82391577395254559468338567665, −5.75226688239127630854676234874, −4.59821644059705405070027823738, −3.92336077756906320269934278479, −2.73575745828592144395738657358, −0.18300310625802154571549899003, 2.50757865896788474581677537599, 3.60682064624161386284187565953, 4.71239604920617240621810309436, 6.07125316137734507505059336069, 6.58626086388303577998344357310, 7.80117635022338425577830061335, 8.265298290147946780825135722465, 9.870606152646370805588985978787, 10.76450215193539395112107457992, 11.78866887661479700132143784416

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