Properties

Label 2-462-33.8-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.368 + 0.929i$
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 + (−0.520 + 1.65i)3-s + (0.309 − 0.951i)4-s + (−1.54 + 2.11i)5-s + (−0.550 − 1.64i)6-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + (−2.45 − 1.71i)9-s − 2.62i·10-s + (0.567 + 3.26i)11-s + (1.41 + 1.00i)12-s + (−0.538 − 0.741i)13-s + (0.951 − 0.309i)14-s + (−2.70 − 3.64i)15-s + (−0.809 − 0.587i)16-s + (−3.44 − 2.50i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.300 + 0.953i)3-s + (0.154 − 0.475i)4-s + (−0.688 + 0.947i)5-s + (−0.224 − 0.670i)6-s + (−0.359 − 0.116i)7-s + (0.109 + 0.336i)8-s + (−0.819 − 0.572i)9-s − 0.828i·10-s + (0.171 + 0.985i)11-s + (0.407 + 0.290i)12-s + (−0.149 − 0.205i)13-s + (0.254 − 0.0825i)14-s + (−0.697 − 0.941i)15-s + (−0.202 − 0.146i)16-s + (−0.835 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.368 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107210 - 0.157851i\)
\(L(\frac12)\) \(\approx\) \(0.107210 - 0.157851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.520 - 1.65i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.567 - 3.26i)T \)
good5 \( 1 + (1.54 - 2.11i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.538 + 0.741i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.44 + 2.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.49 - 0.485i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.00iT - 23T^{2} \)
29 \( 1 + (-2.38 + 7.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.41 - 1.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.04 - 6.30i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.42 - 4.39i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 + (5.43 - 1.76i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.38 + 8.79i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.219 - 0.0714i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.88 - 10.8i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.86T + 67T^{2} \)
71 \( 1 + (4.47 - 6.15i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (13.6 + 4.42i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.66 + 2.28i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.4 - 8.33i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.91iT - 89T^{2} \)
97 \( 1 + (13.7 - 10.0i)T + (29.9 - 92.2i)T^{2} \)
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

−11.42494153487988109890405343118, −10.55460301837333919605625847007, −9.972500788631968761150262191242, −9.125103069668166102929528452223, −8.048581142872026431926792308843, −6.98452768022986256570551636017, −6.36310735651561567313774436071, −4.95687832683503733120842917491, −3.97615536895392892993047574285, −2.67296667782127257530576585078, 0.14652690082289812741936491721, 1.56272012990336296923870111805, 3.13910958631081537996747755355, 4.52566433610459196442642459824, 5.87240722471618961811564059902, 6.84671042401393148642910220603, 7.907350301405568229121424572082, 8.612754233742983097403632510446, 9.224331494463157054608485623527, 10.77438361894008915185788366433

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