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Properties

Label 1-463-463.209-r0-0-0
Degree $1$
Conductor $463$
Sign $0.509 + 0.860i$
Analytic cond. $2.15016$
Root an. cond. $2.15016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.557 − 0.830i)2-s + (−0.301 − 0.953i)3-s + (−0.377 − 0.925i)4-s + (−0.523 − 0.852i)5-s + (−0.959 − 0.281i)6-s + (−0.301 + 0.953i)7-s + (−0.979 − 0.202i)8-s + (−0.818 + 0.574i)9-s + (−0.999 − 0.0407i)10-s + (0.557 + 0.830i)11-s + (−0.768 + 0.639i)12-s + (−0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.654 + 0.755i)15-s + (−0.714 + 0.699i)16-s + (−0.862 − 0.505i)17-s + ⋯
L(s)  = 1  + (0.557 − 0.830i)2-s + (−0.301 − 0.953i)3-s + (−0.377 − 0.925i)4-s + (−0.523 − 0.852i)5-s + (−0.959 − 0.281i)6-s + (−0.301 + 0.953i)7-s + (−0.979 − 0.202i)8-s + (−0.818 + 0.574i)9-s + (−0.999 − 0.0407i)10-s + (0.557 + 0.830i)11-s + (−0.768 + 0.639i)12-s + (−0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.654 + 0.755i)15-s + (−0.714 + 0.699i)16-s + (−0.862 − 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(463\)
Sign: $0.509 + 0.860i$
Analytic conductor: \(2.15016\)
Root analytic conductor: \(2.15016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{463} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 463,\ (0:\ ),\ 0.509 + 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05289804420 + 0.03016155434i\)
\(L(\frac12)\) \(\approx\) \(-0.05289804420 + 0.03016155434i\)
\(L(1)\) \(\approx\) \(0.5182157219 - 0.5419829283i\)
\(L(1)\) \(\approx\) \(0.5182157219 - 0.5419829283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad463 \( 1 \)
good2 \( 1 + (0.557 - 0.830i)T \)
3 \( 1 + (-0.301 - 0.953i)T \)
5 \( 1 + (-0.523 - 0.852i)T \)
7 \( 1 + (-0.301 + 0.953i)T \)
11 \( 1 + (0.557 + 0.830i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.862 - 0.505i)T \)
19 \( 1 + (0.917 + 0.396i)T \)
23 \( 1 + (-0.591 + 0.806i)T \)
29 \( 1 + (-0.714 - 0.699i)T \)
31 \( 1 + (-0.992 + 0.122i)T \)
37 \( 1 + (-0.452 - 0.891i)T \)
41 \( 1 + (0.101 + 0.994i)T \)
43 \( 1 + (-0.591 + 0.806i)T \)
47 \( 1 + (0.262 - 0.965i)T \)
53 \( 1 + (-0.0611 - 0.998i)T \)
59 \( 1 + (0.685 - 0.728i)T \)
61 \( 1 + (0.262 - 0.965i)T \)
67 \( 1 + (0.0203 + 0.999i)T \)
71 \( 1 + (0.685 + 0.728i)T \)
73 \( 1 + (0.182 - 0.983i)T \)
79 \( 1 + (-0.768 - 0.639i)T \)
83 \( 1 + (-0.862 + 0.505i)T \)
89 \( 1 + (-0.862 - 0.505i)T \)
97 \( 1 + (-0.523 + 0.852i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.09086111080146518158262974592, −23.96386242031483001910112112599, −22.60077304214135695273652966095, −22.297084104841351618240928916294, −21.73493257020816613557335626350, −20.42071937227658413506101007936, −19.68939050312986050730784711281, −18.42556560727744521697907603379, −17.32565768249570667424445599723, −16.68601855624092602925661749972, −15.97342396473420810192925607026, −15.122170656407526745293385701024, −14.31457449978525657067412279674, −13.76076282840089662056941585541, −12.32703792898529936346410955799, −11.39483318550439395826015228948, −10.64012219359344086002361356522, −9.502116022925746441694006535047, −8.51816734827449330946847831820, −7.24456115903180826774665961103, −6.65104026284952618720277785585, −5.58809753893452534890271772416, −4.3412053503108269727448412087, −3.798069860277819391034164012793, −2.87827189347091012473613649642, 0.02821806821987602314649100213, 1.569200182092021540688056436686, 2.389482897713879534623296116474, 3.70607983836707983722283245765, 5.03939582060005406698826552292, 5.55524678108045851262273013724, 6.82615870808707011610852795350, 7.92347032031583251882703957679, 9.128361172643474036147088204923, 9.78388020475785933255487116309, 11.53903862737167249233396336181, 11.74405374686742176380160708074, 12.68494102399926709519490837613, 13.082471798316500177980647280258, 14.32011365656692429817342791881, 15.22748893275067121062257732488, 16.199001255549922072833564795186, 17.50478980692339490197243946849, 18.160726831590951298521463434489, 19.14653496199685110269322618486, 19.93006538542016914573150556770, 20.2563765953838887370957942792, 21.66025518317365381307281102812, 22.4790295165434380307854581122, 22.99148264984289022761506439638

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