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Properties

Label 2-462-231.32-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.321 - 0.947i$
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 + (−1.64 + 0.543i)3-s + (−0.499 + 0.866i)4-s + (1.63 − 0.941i)5-s + (1.29 + 1.15i)6-s + (−1.76 + 1.96i)7-s + 0.999·8-s + (2.40 − 1.78i)9-s + (−1.63 − 0.941i)10-s + (−3.18 + 0.913i)11-s + (0.351 − 1.69i)12-s − 1.92i·13-s + (2.58 + 0.546i)14-s + (−2.16 + 2.43i)15-s + (−0.5 − 0.866i)16-s + (−1.83 + 3.17i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.949 + 0.313i)3-s + (−0.249 + 0.433i)4-s + (0.729 − 0.421i)5-s + (0.527 + 0.470i)6-s + (−0.668 + 0.744i)7-s + 0.353·8-s + (0.802 − 0.596i)9-s + (−0.515 − 0.297i)10-s + (−0.961 + 0.275i)11-s + (0.101 − 0.489i)12-s − 0.534i·13-s + (0.691 + 0.145i)14-s + (−0.560 + 0.628i)15-s + (−0.125 − 0.216i)16-s + (−0.443 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.321 - 0.947i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.321 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205100 + 0.286136i\)
\(L(\frac12)\) \(\approx\) \(0.205100 + 0.286136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.543i)T \)
7 \( 1 + (1.76 - 1.96i)T \)
11 \( 1 + (3.18 - 0.913i)T \)
good5 \( 1 + (-1.63 + 0.941i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 1.92iT - 13T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.71 + 2.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.69T + 29T^{2} \)
31 \( 1 + (5.11 - 8.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.15 - 7.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.35T + 41T^{2} \)
43 \( 1 - 6.75iT - 43T^{2} \)
47 \( 1 + (4.43 - 2.55i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.07 - 1.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.50 + 5.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.0 - 5.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.56 + 9.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 + (1.82 + 1.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.73 + 5.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.34T + 83T^{2} \)
89 \( 1 + (10.7 - 6.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.0299T + 97T^{2} \)
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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

−11.01348091830128888647270391024, −10.56776945897568826554426461754, −9.583891012862312377418864893585, −9.061226806090452046067573418581, −7.80350295142354550702240894737, −6.43067218893002858144748992660, −5.60477535781721291713467613184, −4.73720665810303375695270736605, −3.23518134471106596907469509388, −1.76304046676233837775306823448, 0.26181643732025153378477381195, 2.23240477747005613844205252624, 4.16672195412938302716686111282, 5.46184007333612552779078071543, 6.12643333565265391205370871364, 7.10594352029951824071028406810, 7.57935551498975756996582554420, 9.211404256415009828793006981991, 9.845152462618778251909614543566, 10.86745876250099084496901571071

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