Properties

Label 2-462-33.8-c1-0-17
Degree $2$
Conductor $462$
Sign $0.416 + 0.909i$
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  + (0.809 − 0.587i)2-s + (1.60 − 0.640i)3-s + (0.309 − 0.951i)4-s + (−0.354 + 0.487i)5-s + (0.925 − 1.46i)6-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + (2.17 − 2.06i)9-s + 0.602i·10-s + (3.17 − 0.964i)11-s + (−0.112 − 1.72i)12-s + (0.688 + 0.947i)13-s + (−0.951 + 0.309i)14-s + (−0.257 + 1.01i)15-s + (−0.809 − 0.587i)16-s + (−4.38 − 3.18i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.929 − 0.370i)3-s + (0.154 − 0.475i)4-s + (−0.158 + 0.217i)5-s + (0.377 − 0.597i)6-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.726 − 0.687i)9-s + 0.190i·10-s + (0.956 − 0.290i)11-s + (−0.0324 − 0.498i)12-s + (0.190 + 0.262i)13-s + (−0.254 + 0.0825i)14-s + (−0.0664 + 0.261i)15-s + (−0.202 − 0.146i)16-s + (−1.06 − 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.416 + 0.909i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.416 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09919 - 1.34737i\)
\(L(\frac12)\) \(\approx\) \(2.09919 - 1.34737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-1.60 + 0.640i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-3.17 + 0.964i)T \)
good5 \( 1 + (0.354 - 0.487i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.688 - 0.947i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.38 + 3.18i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.55 + 1.15i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.43iT - 23T^{2} \)
29 \( 1 + (-2.42 + 7.46i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.43 - 6.13i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.74 - 11.5i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.34 - 7.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.23iT - 43T^{2} \)
47 \( 1 + (9.20 - 2.99i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.24 + 7.21i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-6.10 - 1.98i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.20 + 5.78i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 + (-2.83 + 3.90i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.39 - 0.454i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.11 - 8.42i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.7 + 7.80i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + (4.11 - 2.98i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.24001440986378625345949145261, −9.744654593393599255513107532908, −9.311708044468869002458340217455, −8.212404586572578414454865593605, −6.97821755583691100213089682208, −6.47929224996743523256244989836, −4.88320846089645117465430142284, −3.69276370874942511885699983147, −2.94114634319039581424985359541, −1.45621144828591038676633354093, 2.10504958661103290019586184495, 3.55833746523533080155276678550, 4.20561336293978899844351222357, 5.43255975325303386945836417086, 6.67245424079582227153947035407, 7.48739188744597824791130979167, 8.673790256858610409010003896527, 9.074943603807557004050604095263, 10.26450045448156645072253770035, 11.19745448949469674839273555915

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