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Properties

Label 1-485-485.443-r0-0-0
Degree $1$
Conductor $485$
Sign $-0.329 - 0.944i$
Analytic cond. $2.25233$
Root an. cond. $2.25233$
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  + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + i·6-s + (0.555 − 0.831i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)12-s + (0.831 − 0.555i)13-s + (−0.831 + 0.555i)14-s + i·16-s + (−0.831 + 0.555i)17-s + (0.923 − 0.382i)18-s + (0.831 − 0.555i)19-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + i·6-s + (0.555 − 0.831i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)12-s + (0.831 − 0.555i)13-s + (−0.831 + 0.555i)14-s + i·16-s + (−0.831 + 0.555i)17-s + (0.923 − 0.382i)18-s + (0.831 − 0.555i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ -0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5132535083 - 0.7226913770i\)
\(L(\frac12)\) \(\approx\) \(0.5132535083 - 0.7226913770i\)
\(L(1)\) \(\approx\) \(0.6314538693 - 0.4011852966i\)
\(L(1)\) \(\approx\) \(0.6314538693 - 0.4011852966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.923 - 0.382i)T \)
3 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.555 - 0.831i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (-0.831 + 0.555i)T \)
19 \( 1 + (0.831 - 0.555i)T \)
23 \( 1 + (0.980 + 0.195i)T \)
29 \( 1 + (-0.195 + 0.980i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (0.980 + 0.195i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
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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

−24.2635854278392281504960454582, −23.087299489893950819743321774772, −22.41119285625199316359540530501, −21.2227812173358080129745376243, −20.71604926007991652671793050311, −19.73941625524725381598368544435, −18.71496057929916947624687801335, −17.83389006043684509463103309392, −17.2878634288788349742355003779, −16.14352377615185448498059953069, −15.77190712222845172245334647092, −14.72511152345129695204209229108, −14.17026878485630974474479687159, −12.3035040276088172213407482528, −11.31702864945035843140166539518, −11.026592826289003808912113127712, −9.536220524228954193176697648329, −9.2039261735778261840343495031, −8.31877166414930552426830786799, −6.97566410071468600444414671478, −6.044657765413369253385322788524, −5.19224645098371410056573958096, −4.07244659987459673135501593566, −2.57229740468388407356157464929, −1.24046662104773298551720587074, 0.869817851089696100110439555548, 1.52109369524818895163978385597, 2.91537588357508444617789327602, 4.11012798935406052254010676031, 5.71339767378441740865609964184, 6.78442913048532427858224338340, 7.42323212044813723220597067727, 8.40577885912118687835084206847, 9.16494132008494921776538330823, 10.67942605067781033613428210039, 11.102315113294420980323489375789, 11.879590581972131130562333271628, 13.04886224213139865935990410538, 13.6293040543923331373791774579, 14.91461273999551573702481207055, 16.17093711305948945465517500957, 17.047814036403470376404421701434, 17.55559170796962876567614352976, 18.31902397258799152867359566910, 19.18575932729296090611169371359, 20.0247304752624859701067593494, 20.52405821337861207383135279766, 21.82866298649486161511222970874, 22.584139879498289933565278378482, 23.771219527268823743862779044228

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