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Properties

Label 2-462-77.10-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.771 - 0.636i$
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  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.606 + 0.350i)5-s − 0.999·6-s + (−1.82 + 1.91i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.350 − 0.606i)10-s + (2.78 + 1.80i)11-s + (0.866 − 0.499i)12-s − 7.03·13-s + (0.629 − 2.56i)14-s − 0.700·15-s + (−0.5 − 0.866i)16-s + (−0.308 + 0.535i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.271 + 0.156i)5-s − 0.408·6-s + (−0.691 + 0.722i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.110 − 0.191i)10-s + (0.838 + 0.545i)11-s + (0.249 − 0.144i)12-s − 1.95·13-s + (0.168 − 0.686i)14-s − 0.180·15-s + (−0.125 − 0.216i)16-s + (−0.0749 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266912 + 0.742483i\)
\(L(\frac12)\) \(\approx\) \(0.266912 + 0.742483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.82 - 1.91i)T \)
11 \( 1 + (-2.78 - 1.80i)T \)
good5 \( 1 + (0.606 - 0.350i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 7.03T + 13T^{2} \)
17 \( 1 + (0.308 - 0.535i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.391 - 0.678i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.63 - 4.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.09iT - 29T^{2} \)
31 \( 1 + (5.35 + 3.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.99 - 3.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.85T + 41T^{2} \)
43 \( 1 + 3.62iT - 43T^{2} \)
47 \( 1 + (2.03 - 1.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.95 - 6.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.25 + 1.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.18 - 7.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.07 - 1.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + (-2.64 + 4.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.75 + 2.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.03T + 83T^{2} \)
89 \( 1 + (-14.0 + 8.10i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−11.36754562393672871103239686568, −10.09526657527100050990229122222, −9.446073922089749470237561003012, −9.004107937629513127929797883370, −7.59870882766104034941505086270, −7.15134177329453624137160624626, −5.88034575608749338715307804707, −4.74843609083530323717746496681, −3.32958550794924235812631679927, −2.08059847909967549534933193485, 0.53711208981894313225437785757, 2.36549013963713380528654773343, 3.51094534076612241702652944301, 4.63523000163807857464402266694, 6.40570721020424639595934531414, 7.21088955969433828218434041917, 7.960744810613905135009168448093, 9.096133775729282180724661866380, 9.655602941084177371475444964326, 10.55188072407130131086823935485

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