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Properties

Label 2-462-7.4-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.991 + 0.126i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s − 4·13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s + (−0.144 − 0.249i)12-s − 1.10·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.117 − 0.204i)18-s + (0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0518918 - 0.817707i\)
\(L(\frac12)\) \(\approx\) \(0.0518918 - 0.817707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−11.67312866896740766943066607040, −10.51363459901816659797583573193, −9.606320892641826848155413594912, −8.934634046754496247601581557331, −7.74792148853239760290927800229, −6.75260617019108600773242741599, −5.82573420156529238865791839443, −4.98890793324664761846083101511, −3.86508632289546169071199358253, −2.59051146885381132687769233625, 0.44328744894148832756782786247, 2.28431823360096414853500230929, 3.47636017692465903494795314282, 4.74805101031080637312037232329, 5.75351273109874385736137096373, 6.92772657536207404805608061791, 7.56197122561348659510900410278, 9.092519478529019722256134575738, 9.760086933343852107114036719694, 10.84288988036191239679969581820

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