Properties

Label 2-462-33.2-c1-0-12
Degree $2$
Conductor $462$
Sign $0.261 + 0.965i$
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  + (−0.309 + 0.951i)2-s + (−1.48 − 0.893i)3-s + (−0.809 − 0.587i)4-s + (0.599 − 0.194i)5-s + (1.30 − 1.13i)6-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s + (1.40 + 2.65i)9-s + 0.630i·10-s + (−2.44 − 2.24i)11-s + (0.674 + 1.59i)12-s + (2.41 + 0.783i)13-s + (−0.587 − 0.809i)14-s + (−1.06 − 0.246i)15-s + (0.309 + 0.951i)16-s + (−1.70 − 5.24i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.856 − 0.516i)3-s + (−0.404 − 0.293i)4-s + (0.268 − 0.0871i)5-s + (0.534 − 0.463i)6-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s + (0.467 + 0.884i)9-s + 0.199i·10-s + (−0.737 − 0.675i)11-s + (0.194 + 0.460i)12-s + (0.669 + 0.217i)13-s + (−0.157 − 0.216i)14-s + (−0.274 − 0.0637i)15-s + (0.0772 + 0.237i)16-s + (−0.413 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505121 - 0.386507i\)
\(L(\frac12)\) \(\approx\) \(0.505121 - 0.386507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (1.48 + 0.893i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (2.44 + 2.24i)T \)
good5 \( 1 + (-0.599 + 0.194i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-2.41 - 0.783i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.70 + 5.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.782 + 1.07i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.39iT - 23T^{2} \)
29 \( 1 + (0.707 + 0.513i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.94 + 9.06i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.24 + 0.908i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.18 + 2.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.14iT - 43T^{2} \)
47 \( 1 + (-1.24 - 1.71i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.81 + 0.915i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.03 - 5.54i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.4 + 3.73i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.01T + 67T^{2} \)
71 \( 1 + (13.8 - 4.51i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.66 + 2.28i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.35 + 3.03i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.56 - 10.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 + (-1.98 + 6.11i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.93211448305504574446047653828, −9.963510249183815062563579229485, −8.928553783387424802322695201974, −8.023615535556245555689923271872, −7.03954304101878586292390176863, −6.15310914529124401321670331329, −5.50007317773395392686884692664, −4.40548975531856438672088277921, −2.42675703403411787091198535777, −0.48128603793928463075387637925, 1.58272666528578837802281911947, 3.37172721278512095215885467980, 4.35771976637251508779388053904, 5.48557280000216974785221463465, 6.42682748904839936117725185540, 7.64743139796753988551616411060, 8.779979792502919744042976800836, 9.885245391992766374020967947677, 10.33012250842664728376279916491, 11.05994631888736035672534312320

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