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Properties

Label 2-462-231.65-c1-0-20
Degree $2$
Conductor $462$
Sign $0.333 + 0.942i$
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 + (0.783 − 1.54i)3-s + (−0.499 − 0.866i)4-s + (3.23 + 1.86i)5-s + (−0.946 − 1.45i)6-s + (−0.830 + 2.51i)7-s − 0.999·8-s + (−1.77 − 2.42i)9-s + (3.23 − 1.86i)10-s + (3.31 − 0.0598i)11-s + (−1.72 + 0.0938i)12-s − 4.64i·13-s + (1.76 + 1.97i)14-s + (5.42 − 3.53i)15-s + (−0.5 + 0.866i)16-s + (−0.460 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.452 − 0.891i)3-s + (−0.249 − 0.433i)4-s + (1.44 + 0.835i)5-s + (−0.386 − 0.592i)6-s + (−0.313 + 0.949i)7-s − 0.353·8-s + (−0.590 − 0.806i)9-s + (1.02 − 0.590i)10-s + (0.999 − 0.0180i)11-s + (−0.499 + 0.0270i)12-s − 1.28i·13-s + (0.470 + 0.527i)14-s + (1.39 − 0.912i)15-s + (−0.125 + 0.216i)16-s + (−0.111 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.333 + 0.942i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86688 - 1.31986i\)
\(L(\frac12)\) \(\approx\) \(1.86688 - 1.31986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.783 + 1.54i)T \)
7 \( 1 + (0.830 - 2.51i)T \)
11 \( 1 + (-3.31 + 0.0598i)T \)
good5 \( 1 + (-3.23 - 1.86i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 + (0.460 + 0.798i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.14 - 2.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.94 + 2.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.73T + 29T^{2} \)
31 \( 1 + (-1.26 - 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.29 - 2.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 + (-4.96 - 2.86i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.30 + 1.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.65 - 4.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.52 + 3.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.390 - 0.676i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (-9.35 + 5.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-14.9 - 8.60i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + (7.44 + 4.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−10.92930639923400599582294749689, −9.826617541165384522992926235619, −9.380338297947859465885597715167, −8.276100216198741013183117506159, −6.94616551575676539041901089283, −6.01219865689677132396527928050, −5.57402820173601771802190397675, −3.41089012797521220951339324656, −2.60490252604242894947118162386, −1.60568710260873232211072906862, 1.84697716488012368390047828348, 3.67819200930034414848064913622, 4.47525286736310996008195320949, 5.49093578665571781364800614413, 6.40578764109515673465366406601, 7.50466551540690374375369130878, 8.859929019232769899276415329308, 9.401098871368359311958260841201, 9.883506436618965761336753473596, 11.13122160292180468037115741052

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