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Properties

Label 2-462-231.146-c1-0-29
Degree $2$
Conductor $462$
Sign $-0.974 + 0.224i$
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  + (−0.587 − 0.809i)2-s + (0.582 − 1.63i)3-s + (−0.309 + 0.951i)4-s + (−0.793 − 0.576i)5-s + (−1.66 + 0.487i)6-s + (0.441 − 2.60i)7-s + (0.951 − 0.309i)8-s + (−2.32 − 1.90i)9-s + 0.980i·10-s + (2.99 − 1.43i)11-s + (1.37 + 1.05i)12-s + (1.73 + 2.38i)13-s + (−2.37 + 1.17i)14-s + (−1.40 + 0.957i)15-s + (−0.809 − 0.587i)16-s + (−6.42 − 4.66i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.336 − 0.941i)3-s + (−0.154 + 0.475i)4-s + (−0.354 − 0.257i)5-s + (−0.678 + 0.198i)6-s + (0.167 − 0.985i)7-s + (0.336 − 0.109i)8-s + (−0.773 − 0.633i)9-s + 0.310i·10-s + (0.901 − 0.432i)11-s + (0.395 + 0.305i)12-s + (0.481 + 0.662i)13-s + (−0.633 + 0.314i)14-s + (−0.362 + 0.247i)15-s + (−0.202 − 0.146i)16-s + (−1.55 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.974 + 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115693 - 1.01585i\)
\(L(\frac12)\) \(\approx\) \(0.115693 - 1.01585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.582 + 1.63i)T \)
7 \( 1 + (-0.441 + 2.60i)T \)
11 \( 1 + (-2.99 + 1.43i)T \)
good5 \( 1 + (0.793 + 0.576i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.73 - 2.38i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.42 + 4.66i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.35 - 0.440i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.16iT - 23T^{2} \)
29 \( 1 + (-3.54 - 1.15i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.63 + 5.00i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.129 + 0.398i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.484 - 1.49i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 + (-0.768 - 2.36i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.386 - 0.532i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.67 - 5.15i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.32 + 8.71i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.50T + 67T^{2} \)
71 \( 1 + (-3.68 + 5.07i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.22 - 1.69i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.4 + 9.04i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.34 + 3.88i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (-5.45 - 7.51i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.98514786616645363330865224627, −9.478963671464065447267714551000, −8.901460983637247734913686376382, −7.950800848981133237669038820454, −7.10857667129489754417457359128, −6.30291261816824636044500629237, −4.45519280753860756865890225827, −3.54409128332889803748762162692, −2.01460200332943845228910029560, −0.70627449801994066382446753434, 2.24287898635009375439009254844, 3.76966079619404177191993228468, 4.75838769652669996505144667194, 5.91288451921857294449576065221, 6.79858696191210342758152900397, 8.241344831295780740313392954597, 8.698052713018927694824580648041, 9.423874716189951825760268532058, 10.59346902485921343567892372007, 11.07168333882647966748962420974

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