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Properties

Label 2-462-33.2-c1-0-4
Degree $2$
Conductor $462$
Sign $0.802 - 0.596i$
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 + (−1.58 + 0.694i)3-s + (−0.809 − 0.587i)4-s + (−1.45 + 0.471i)5-s + (0.170 + 1.72i)6-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s + (2.03 − 2.20i)9-s + 1.52i·10-s + (−3.21 + 0.816i)11-s + (1.69 + 0.370i)12-s + (3.83 + 1.24i)13-s + (−0.587 − 0.809i)14-s + (1.97 − 1.75i)15-s + (0.309 + 0.951i)16-s + (1.92 + 5.93i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.915 + 0.401i)3-s + (−0.404 − 0.293i)4-s + (−0.648 + 0.210i)5-s + (0.0696 + 0.703i)6-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s + (0.678 − 0.735i)9-s + 0.482i·10-s + (−0.969 + 0.246i)11-s + (0.488 + 0.106i)12-s + (1.06 + 0.345i)13-s + (−0.157 − 0.216i)14-s + (0.509 − 0.453i)15-s + (0.0772 + 0.237i)16-s + (0.467 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.802 - 0.596i$
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: \((2,\ 462,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821910 + 0.272154i\)
\(L(\frac12)\) \(\approx\) \(0.821910 + 0.272154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.694i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (3.21 - 0.816i)T \)
good5 \( 1 + (1.45 - 0.471i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-3.83 - 1.24i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.92 - 5.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.38 - 6.03i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.184iT - 23T^{2} \)
29 \( 1 + (-2.12 - 1.54i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.712 + 2.19i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.41 - 2.48i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.03 + 1.47i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.14iT - 43T^{2} \)
47 \( 1 + (6.08 + 8.37i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.46 - 0.477i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.34 - 4.60i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-6.79 + 2.20i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.875T + 67T^{2} \)
71 \( 1 + (5.30 - 1.72i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.87 - 6.70i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.19 + 2.98i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.95 - 6.02i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.2iT - 89T^{2} \)
97 \( 1 + (-4.34 + 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.20226156676947065396143490433, −10.37376249830088401701559274832, −9.875392276271935991956603883986, −8.417497238153447637339373133866, −7.55046195005410601413481406068, −6.17798936573597836938853299765, −5.37319799906360026391062046103, −4.14350965608525653851738599517, −3.48665779107516785535453948720, −1.39886095789410429229626784754, 0.63445649800832632411040465324, 2.97254284447124026170213898255, 4.59723738416255459985691341651, 5.29855585302358267415497215164, 6.19402856041482634305930608989, 7.36900636512569195170020600640, 7.87227653146172006790757084672, 8.943609952553785363828179645680, 10.15103718223574370504539081786, 11.34124307156846112402782757979

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