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Properties

Label 2-462-11.9-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.704 + 0.710i$
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 + (0.330 + 1.01i)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 − 1.07·10-s + (−3.25 − 0.654i)11-s + 0.999·12-s + (−2.12 + 6.53i)13-s + (0.809 − 0.587i)14-s + (−0.866 − 0.629i)15-s + (0.309 + 0.951i)16-s + (−2.18 − 6.73i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.147 + 0.455i)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 − 0.338·10-s + (−0.980 − 0.197i)11-s + 0.288·12-s + (−0.589 + 1.81i)13-s + (0.216 − 0.157i)14-s + (−0.223 − 0.162i)15-s + (0.0772 + 0.237i)16-s + (−0.530 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.704 + 0.710i$
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: \((2,\ 462,\ (\ :1/2),\ -0.704 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0748537 - 0.179669i\)
\(L(\frac12)\) \(\approx\) \(0.0748537 - 0.179669i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 + 0.654i)T \)
good5 \( 1 + (-0.330 - 1.01i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (2.12 - 6.53i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.18 + 6.73i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.70 - 3.41i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
29 \( 1 + (3.14 + 2.28i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.10 + 3.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.63 + 1.91i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.92 + 2.84i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.20T + 43T^{2} \)
47 \( 1 + (8.22 - 5.97i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.18 - 9.79i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.410 + 0.298i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.47 - 7.60i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.79T + 67T^{2} \)
71 \( 1 + (4.39 + 13.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.4 - 7.56i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.77 - 8.54i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.553 + 1.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.23T + 89T^{2} \)
97 \( 1 + (5.17 - 15.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.43029528094329097356334804007, −10.60991837355032139473967733047, −9.735512618573744278821653512011, −9.076069908475903638230111727712, −7.80744872151480049347368858914, −6.86987981802481459951085368811, −6.22679473311011927406238456836, −5.00226051292183421695011199761, −4.13696369672813120520699858455, −2.39997074364338936707924314890, 0.12765300321659256116737686567, 1.95388914810413078260587428648, 3.21783210837416938015018182641, 4.78630795755661295906857298250, 5.55707116442635011098287441388, 6.75873815377856571911645776398, 8.058801237059852877738856822299, 8.557258409756849478807312879129, 9.917359005759738885281346874850, 10.50930525301264993406018741600

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