Properties

Label 2-462-231.95-c1-0-18
Degree $2$
Conductor $462$
Sign $0.842 - 0.539i$
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  + (−0.104 + 0.994i)2-s + (1.67 − 0.438i)3-s + (−0.978 − 0.207i)4-s + (−0.684 + 1.53i)5-s + (0.261 + 1.71i)6-s + (1.54 − 2.14i)7-s + (0.309 − 0.951i)8-s + (2.61 − 1.47i)9-s + (−1.45 − 0.841i)10-s + (2.83 − 1.72i)11-s + (−1.73 + 0.0808i)12-s + (1.78 − 2.46i)13-s + (1.97 + 1.76i)14-s + (−0.472 + 2.87i)15-s + (0.913 + 0.406i)16-s + (0.447 + 4.26i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (0.967 − 0.253i)3-s + (−0.489 − 0.103i)4-s + (−0.306 + 0.687i)5-s + (0.106 + 0.699i)6-s + (0.585 − 0.810i)7-s + (0.109 − 0.336i)8-s + (0.871 − 0.490i)9-s + (−0.461 − 0.266i)10-s + (0.853 − 0.521i)11-s + (−0.499 + 0.0233i)12-s + (0.495 − 0.682i)13-s + (0.526 + 0.471i)14-s + (−0.121 + 0.743i)15-s + (0.228 + 0.101i)16-s + (0.108 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.842 - 0.539i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.842 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80619 + 0.528513i\)
\(L(\frac12)\) \(\approx\) \(1.80619 + 0.528513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (-1.67 + 0.438i)T \)
7 \( 1 + (-1.54 + 2.14i)T \)
11 \( 1 + (-2.83 + 1.72i)T \)
good5 \( 1 + (0.684 - 1.53i)T + (-3.34 - 3.71i)T^{2} \)
13 \( 1 + (-1.78 + 2.46i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.447 - 4.26i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (0.153 + 0.720i)T + (-17.3 + 7.72i)T^{2} \)
23 \( 1 + (6.96 - 4.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.76 - 8.50i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.99 - 2.66i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-4.23 + 4.70i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (-0.907 + 2.79i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.89iT - 43T^{2} \)
47 \( 1 + (2.33 + 10.9i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (0.415 + 0.932i)T + (-35.4 + 39.3i)T^{2} \)
59 \( 1 + (1.40 - 6.60i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (2.09 - 4.70i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (2.70 - 4.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.86 + 13.5i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.10 - 14.5i)T + (-66.6 - 29.6i)T^{2} \)
79 \( 1 + (-1.58 - 0.166i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (-0.791 + 0.575i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.79 + 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.6 + 8.50i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.86040933679157368672483535800, −10.25820822892755580767158029724, −8.975859753611495164392074622826, −8.325088842136045538012468973683, −7.44594334120880009888083435892, −6.84343300294713370685618545825, −5.66054256604076445130622759946, −3.99564350526585905950959912288, −3.46143656668680350192756434364, −1.47330711844501194627371593768, 1.58554170702075303537144518777, 2.68993879153957861046313216032, 4.22461622295898338427176977829, 4.62293086819465122046210023330, 6.25582778525118516554002324791, 7.83317537157284705185498270689, 8.385466903991720033619670249989, 9.347884361277990110417785241789, 9.689945377590412510668957943958, 11.10001406786558961730876638102

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