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Properties

Label 2-462-77.41-c1-0-4
Degree $2$
Conductor $462$
Sign $0.976 - 0.216i$
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 + (−0.884 + 1.21i)5-s + (0.809 + 0.587i)6-s + (−1.60 − 2.10i)7-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + 1.50·10-s + (2.41 − 2.27i)11-s − i·12-s + (−2.55 + 1.85i)13-s + (−0.763 + 2.53i)14-s + (0.465 − 1.43i)15-s + (−0.809 − 0.587i)16-s + (3.63 + 2.64i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.395 + 0.544i)5-s + (0.330 + 0.239i)6-s + (−0.604 − 0.796i)7-s + (0.336 − 0.109i)8-s + (0.269 − 0.195i)9-s + 0.475·10-s + (0.727 − 0.685i)11-s − 0.288i·12-s + (−0.707 + 0.514i)13-s + (−0.204 + 0.677i)14-s + (0.120 − 0.369i)15-s + (−0.202 − 0.146i)16-s + (0.882 + 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.976 - 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793726 + 0.0868776i\)
\(L(\frac12)\) \(\approx\) \(0.793726 + 0.0868776i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 2.10i)T \)
11 \( 1 + (-2.41 + 2.27i)T \)
good5 \( 1 + (0.884 - 1.21i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.55 - 1.85i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.63 - 2.64i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.96 - 6.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.51T + 23T^{2} \)
29 \( 1 + (-0.814 - 0.264i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.69 - 5.08i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.85 - 5.72i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.11 + 9.59i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.65iT - 43T^{2} \)
47 \( 1 + (-1.41 + 0.460i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.331 + 0.240i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.04 - 1.63i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.80 - 4.21i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (-11.6 - 8.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.34 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.69 - 10.5i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.08 + 1.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.90iT - 89T^{2} \)
97 \( 1 + (9.52 + 13.1i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.89631554070534220484324745947, −10.36057418286913064760920099571, −9.560991362407263656031938362954, −8.503722849150994778789662761177, −7.26697020501013773751702262696, −6.70492651976684320441854261552, −5.33826881122325522614022159370, −3.87352772185538460325763782320, −3.25546516003530730640721503688, −1.15607180551800653789478427842, 0.77361255874662371745337850754, 2.80852033275305422440789082485, 4.67127331758322187712216563532, 5.31389714118262674017550335303, 6.53278558870682895344238587697, 7.23262789473257136135059445453, 8.257446107208586796089706287922, 9.392528287862991152813457880545, 9.701282837276893508443606484928, 11.11084344322307515038431504204

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