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Properties

Label 2-462-231.65-c1-0-9
Degree $2$
Conductor $462$
Sign $-0.465 - 0.885i$
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  + (−0.5 + 0.866i)2-s + (1.32 + 1.11i)3-s + (−0.499 − 0.866i)4-s + (1.38 + 0.800i)5-s + (−1.62 + 0.591i)6-s + (−0.229 + 2.63i)7-s + 0.999·8-s + (0.519 + 2.95i)9-s + (−1.38 + 0.800i)10-s + (−0.523 − 3.27i)11-s + (0.301 − 1.70i)12-s + 2.95i·13-s + (−2.16 − 1.51i)14-s + (0.948 + 2.60i)15-s + (−0.5 + 0.866i)16-s + (−1.22 − 2.11i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.765 + 0.642i)3-s + (−0.249 − 0.433i)4-s + (0.620 + 0.358i)5-s + (−0.664 + 0.241i)6-s + (−0.0867 + 0.996i)7-s + 0.353·8-s + (0.173 + 0.984i)9-s + (−0.438 + 0.253i)10-s + (−0.157 − 0.987i)11-s + (0.0869 − 0.492i)12-s + 0.820i·13-s + (−0.579 − 0.405i)14-s + (0.244 + 0.673i)15-s + (−0.125 + 0.216i)16-s + (−0.296 − 0.512i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.465 - 0.885i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.465 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808709 + 1.33865i\)
\(L(\frac12)\) \(\approx\) \(0.808709 + 1.33865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.32 - 1.11i)T \)
7 \( 1 + (0.229 - 2.63i)T \)
11 \( 1 + (0.523 + 3.27i)T \)
good5 \( 1 + (-1.38 - 0.800i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.95iT - 13T^{2} \)
17 \( 1 + (1.22 + 2.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.26 - 1.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.475 - 0.274i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.955T + 29T^{2} \)
31 \( 1 + (-2.87 - 4.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.653 + 1.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.94T + 41T^{2} \)
43 \( 1 + 2.42iT - 43T^{2} \)
47 \( 1 + (5.63 + 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.07 + 5.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.3 - 6.57i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.48 - 1.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.30 + 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.37iT - 71T^{2} \)
73 \( 1 + (-10.0 + 5.77i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.5 - 6.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.91T + 83T^{2} \)
89 \( 1 + (-5.89 - 3.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−11.07911327207323253893862997519, −10.15844183844809120965057481855, −9.340666496139912809579462252844, −8.781925810064539661343214163637, −7.941906083599164478834006305352, −6.68730604670198225624538223100, −5.75568496123230278863890826008, −4.80424546370417877986298116403, −3.28658698736647879118472602095, −2.13772836589658498366790886430, 1.08359093908707417051348170062, 2.29656435965686436787114170495, 3.54620414244973158708255191334, 4.73654107749067087277879714117, 6.27489266562584138757019959307, 7.41897182538536590888577961355, 7.971161363487182874783615057312, 9.144663490538375542327834194671, 9.803174790218251759365500728966, 10.52962415970279705963932571485

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