↑

Properties

Label 2-462-33.29-c1-0-22
Degree $2$
Conductor $462$
Sign $0.246 + 0.969i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.794 − 1.53i)3-s + (0.309 + 0.951i)4-s + (−1.43 − 1.97i)5-s + (1.54 − 0.778i)6-s + (−0.951 + 0.309i)7-s + (−0.309 + 0.951i)8-s + (−1.73 − 2.44i)9-s − 2.44i·10-s + (−1.53 − 2.93i)11-s + (1.70 + 0.280i)12-s + (4.02 − 5.54i)13-s + (−0.951 − 0.309i)14-s + (−4.18 + 0.640i)15-s + (−0.809 + 0.587i)16-s + (−1.36 + 0.993i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.458 − 0.888i)3-s + (0.154 + 0.475i)4-s + (−0.642 − 0.884i)5-s + (0.631 − 0.317i)6-s + (−0.359 + 0.116i)7-s + (−0.109 + 0.336i)8-s + (−0.579 − 0.815i)9-s − 0.773i·10-s + (−0.462 − 0.886i)11-s + (0.493 + 0.0808i)12-s + (1.11 − 1.53i)13-s + (−0.254 − 0.0825i)14-s + (−1.08 + 0.165i)15-s + (−0.202 + 0.146i)16-s + (−0.331 + 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43051 - 1.11185i\)
\(L(\frac12)\) \(\approx\) \(1.43051 - 1.11185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.794 + 1.53i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (1.53 + 2.93i)T \)
good5 \( 1 + (1.43 + 1.97i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.02 + 5.54i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.36 - 0.993i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.15 - 1.99i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.69iT - 23T^{2} \)
29 \( 1 + (1.73 + 5.32i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.638 + 0.463i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.12 - 6.52i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.79 - 8.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.16iT - 43T^{2} \)
47 \( 1 + (-4.81 - 1.56i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.765 - 1.05i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-13.0 + 4.22i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.87 - 8.09i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 + (-4.46 - 6.14i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.108 + 0.0352i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.14 - 8.45i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.09 + 1.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 + (-0.828 - 0.601i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30049029206741602566605755469, −9.807237988767547315249352942183, −8.456061269853348191241208235151, −8.221446158437638262906767843723, −7.33076825306024697729501090913, −5.97656883852983334019379559462, −5.45552899817502009963538956920, −3.77060904978700832656768404597, −3.01228719946062308676358508824, −0.935905635087156379101505909664, 2.32129844234829691534205687814, 3.44951955223347322917494482974, 4.16908980903719064990953159846, 5.22376075370161399338095864791, 6.67573715128177031361780441572, 7.40233591380079474370350857726, 8.826857138457214242578791520911, 9.554475110916109206108766733108, 10.57722389332090156879444825415, 11.11735953035139331674462008330

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