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Properties

Label 2-462-11.9-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.780 - 0.625i$
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  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.427 − 1.31i)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 + 1.38·10-s + (−0.309 + 3.30i)11-s + 0.999·12-s + (−0.381 + 1.17i)13-s + (−0.809 + 0.587i)14-s + (1.11 + 0.812i)15-s + (0.309 + 0.951i)16-s + (2.35 + 7.24i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.190 − 0.587i)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 + 0.437·10-s + (−0.0931 + 0.995i)11-s + 0.288·12-s + (−0.105 + 0.326i)13-s + (−0.216 + 0.157i)14-s + (0.288 + 0.209i)15-s + (0.0772 + 0.237i)16-s + (0.570 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250964 + 0.714157i\)
\(L(\frac12)\) \(\approx\) \(0.250964 + 0.714157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 3.30i)T \)
good5 \( 1 + (0.427 + 1.31i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.381 - 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.35 - 7.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.30 - 2.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 9.32T + 23T^{2} \)
29 \( 1 + (-1 - 0.726i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.336 + 1.03i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.97 - 5.06i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.59 - 6.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (8.23 - 5.98i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.76 + 8.50i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.23 - 5.25i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.85 + 8.78i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + (-0.472 - 1.45i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.47 - 4.70i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.38 - 7.33i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.70 - 14.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 + (-3.47 + 10.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.39731926223385019909563611951, −10.15756854108297848718338722931, −9.800366238764075969424072279156, −8.352205833695885609840542014302, −8.076937112992542481300009992854, −6.60946481345773416199663192588, −5.86265304610921556627310879370, −4.71905236663916882994324968400, −4.02556044693945944124665351080, −1.71994509486623232268839802055, 0.54236191123585948230760585494, 2.37534112523506956917378762663, 3.53505822108427828487962040709, 4.87194595095778443472059548526, 5.97117666017321595232030540762, 7.15247380164770849142465848160, 7.931894702525411979018098363031, 8.958705185671186025812355851062, 10.11835628363836559146574155700, 10.77510965548990309119660977936

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