↑

Properties

Label 2-462-11.5-c1-0-9
Degree $2$
Conductor $462$
Sign $-0.0632 + 0.997i$
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  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (1.14 − 3.53i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s − 3.71·10-s + (2.64 + 1.99i)11-s − 0.999·12-s + (−1.78 − 5.49i)13-s + (−0.809 − 0.587i)14-s + (3.00 − 2.18i)15-s + (0.309 − 0.951i)16-s + (−1.5 + 4.61i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.513 − 1.58i)5-s + (0.126 − 0.388i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s − 1.17·10-s + (0.798 + 0.601i)11-s − 0.288·12-s + (−0.495 − 1.52i)13-s + (−0.216 − 0.157i)14-s + (0.776 − 0.564i)15-s + (0.0772 − 0.237i)16-s + (−0.363 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0632 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0632 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.0632 + 0.997i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.0632 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06303 - 1.13254i\)
\(L(\frac12)\) \(\approx\) \(1.06303 - 1.13254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-2.64 - 1.99i)T \)
good5 \( 1 + (-1.14 + 3.53i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.78 + 5.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.5 - 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.74 + 1.99i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.75T + 23T^{2} \)
29 \( 1 + (-4.94 + 3.59i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.297 - 0.915i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.22 + 2.34i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.07 + 6.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.48T + 43T^{2} \)
47 \( 1 + (-3.32 - 2.41i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.20 - 9.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.94 + 5.04i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.37 - 10.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.45T + 67T^{2} \)
71 \( 1 + (4.53 - 13.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (10.5 - 7.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.75 + 5.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.61 - 8.04i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + (0.721 + 2.22i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.49704126671177355585901329351, −10.00254518739099095697085185994, −8.885029007671712246494240420557, −8.597528760272090102561226110864, −7.50098928309440603010778628398, −5.81801003408300808300248049530, −4.74048815192448830740939660576, −4.05881867092767759865561908342, −2.39191520851080517764326881899, −1.09116800183065944894519188084, 1.95430171210312484802385996826, 3.16359954493363615412387091370, 4.58732336498438837829000226801, 6.14679194815171303812159559680, 6.73051190452620773520577928926, 7.34557114889419485929548281434, 8.616031173889075892471450906131, 9.341565372353544348322857242972, 10.21490818455384051382640198364, 11.28190640768429976435424685046

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