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Properties

Label 2-462-77.62-c1-0-11
Degree $2$
Conductor $462$
Sign $0.0442 + 0.999i$
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 + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.946 − 1.30i)5-s + (−0.809 + 0.587i)6-s + (−2.49 − 0.877i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + 1.60·10-s + (−2.29 − 2.39i)11-s − i·12-s + (−1.19 − 0.865i)13-s + (2.17 − 1.50i)14-s + (−0.497 − 1.53i)15-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.224i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 − 0.475i)4-s + (−0.423 − 0.582i)5-s + (−0.330 + 0.239i)6-s + (−0.943 − 0.331i)7-s + (0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + 0.509·10-s + (−0.693 − 0.720i)11-s − 0.288i·12-s + (−0.330 − 0.240i)13-s + (0.581 − 0.401i)14-s + (−0.128 − 0.395i)15-s + (−0.202 + 0.146i)16-s + (−0.0750 + 0.0545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0442 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0442 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0442 + 0.999i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0442 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494760 - 0.473322i\)
\(L(\frac12)\) \(\approx\) \(0.494760 - 0.473322i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (2.49 + 0.877i)T \)
11 \( 1 + (2.29 + 2.39i)T \)
good5 \( 1 + (0.946 + 1.30i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.19 + 0.865i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.309 - 0.224i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.11 + 6.49i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 + (8.63 - 2.80i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.73 + 6.51i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.778 - 2.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.174 + 0.537i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + (-3.78 - 1.22i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.56 + 1.13i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.74 + 1.54i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.93 - 2.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + (-1.45 + 1.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.29 + 13.2i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.135 - 0.186i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.4 - 8.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 7.19iT - 89T^{2} \)
97 \( 1 + (9.25 - 12.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.58321106462718603952431155451, −9.695990961320969787076990919137, −9.013332803598381948236317576703, −8.084472905208150873910344887087, −7.39456903355180957201426531884, −6.26712403287393237300453655630, −5.13947165458865795889469863609, −3.99023647693645210751018516542, −2.70431765164906685322514571827, −0.43367235219879630439135193198, 2.03947417497434375406848565991, 3.12654398098606372548952359751, 4.02462032072507487286818404015, 5.67455154302911643660840965739, 7.01112525810542028702477794694, 7.65079956502425570216306317669, 8.620189691555905358968466068200, 9.791493146406625040615699467670, 10.04120229581503584741500314700, 11.23731312761054771164817751484

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