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Properties

Label 2-462-77.54-c1-0-11
Degree $2$
Conductor $462$
Sign $-0.670 + 0.741i$
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 + (0.725 + 0.418i)5-s + 0.999·6-s + (−2.44 − 1.02i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.418 − 0.725i)10-s + (−3.15 + 1.01i)11-s + (−0.866 − 0.499i)12-s + 2.59·13-s + (1.60 + 2.10i)14-s − 0.837·15-s + (−0.5 + 0.866i)16-s + (−2.98 − 5.17i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.324 + 0.187i)5-s + 0.408·6-s + (−0.922 − 0.386i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.132 − 0.229i)10-s + (−0.951 + 0.306i)11-s + (−0.249 − 0.144i)12-s + 0.719·13-s + (0.428 + 0.562i)14-s − 0.216·15-s + (−0.125 + 0.216i)16-s + (−0.724 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.670 + 0.741i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159234 - 0.358668i\)
\(L(\frac12)\) \(\approx\) \(0.159234 - 0.358668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.02i)T \)
11 \( 1 + (3.15 - 1.01i)T \)
good5 \( 1 + (-0.725 - 0.418i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 + (2.98 + 5.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.55 + 2.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.43 - 2.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.38iT - 29T^{2} \)
31 \( 1 + (0.913 - 0.527i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.49 + 9.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 1.27iT - 43T^{2} \)
47 \( 1 + (10.6 + 6.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.58 + 4.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.38 - 4.84i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.03 - 3.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.51 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (-4.95 - 8.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.7 - 6.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.99T + 83T^{2} \)
89 \( 1 + (-7.28 - 4.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.786iT - 97T^{2} \)
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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

−10.66509707840222996447491984809, −9.789785081497822393219453383406, −9.344893382220390019884862746613, −8.019556177040963341138416460517, −7.00612214715871424965889008258, −6.18858927099357310234017813668, −4.94576909095005004517185577590, −3.61239914349499310135602800245, −2.39165465111588241835397365918, −0.30251181642814331507091943220, 1.68395959354423926997103661904, 3.30040297975469260744357945625, 5.04207212894550137032625192358, 6.08707655343468923103196088214, 6.50477503370422852467505985095, 7.87056803954128826266946248160, 8.588553073231831682025354439875, 9.616854795064716382303162445066, 10.42273819408512151977679512653, 11.14267354231440052726159185077

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