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Properties

Label 2-462-11.5-c1-0-4
Degree $2$
Conductor $462$
Sign $0.847 + 0.530i$
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 + (3.01 − 1.38i)11-s + 0.999·12-s + (0.269 + 0.830i)13-s + (0.809 + 0.587i)14-s + (0.866 − 0.629i)15-s + (0.309 − 0.951i)16-s + (−0.0475 + 0.146i)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 + (0.909 − 0.416i)11-s + 0.288·12-s + (0.0748 + 0.230i)13-s + (0.216 + 0.157i)14-s + (0.223 − 0.162i)15-s + (0.0772 − 0.237i)16-s + (−0.0115 + 0.0355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.847 + 0.530i$
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: \((2,\ 462,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998001 - 0.286253i\)
\(L(\frac12)\) \(\approx\) \(0.998001 - 0.286253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.01 + 1.38i)T \)
good5 \( 1 + (0.330 - 1.01i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.269 - 0.830i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.0475 - 0.146i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.70 - 2.69i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.77T + 23T^{2} \)
29 \( 1 + (-6.99 + 5.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.10 + 6.48i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.83 - 5.69i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.68 + 1.22i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.74T + 43T^{2} \)
47 \( 1 + (-8.84 - 6.42i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.19 + 3.68i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.26 + 4.55i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.79 - 11.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + (-0.393 + 1.21i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.32 - 3.87i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.225 + 0.692i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.35 - 10.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + (-1.08 - 3.34i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.14600809306231054240360047301, −10.19375408580266761350354413853, −9.304132245701002833066804091217, −8.401087523097966295506781624621, −7.21831716699942702068970185415, −6.40663089737261320787897275432, −5.24889324202969703925302472364, −3.88307925096935197776707105994, −2.79417839993772989984011628802, −1.15386876949962160614081465805, 0.999587154574059459307954624636, 3.37993851486616771145193616861, 4.67626630969032733324135923216, 5.34464625317727128990128981881, 6.72456589486864976695897549188, 7.13898134261016300572753896050, 8.668085517494732655726984834765, 9.120268413542413626142147284956, 10.18023626337031285513775092850, 10.95619421757354876237253038188

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