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Properties

Label 1-1476-1476.443-r0-0-0
Degree $1$
Conductor $1476$
Sign $-0.545 + 0.838i$
Analytic cond. $6.85451$
Root an. cond. $6.85451$
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  + (−0.104 − 0.994i)5-s + (−0.207 − 0.978i)7-s + (0.994 + 0.104i)11-s + (−0.207 + 0.978i)13-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)19-s + (−0.978 − 0.207i)23-s + (−0.978 + 0.207i)25-s + (−0.406 + 0.913i)29-s + (−0.913 + 0.406i)31-s + (−0.951 + 0.309i)35-s + (−0.809 + 0.587i)37-s + (0.669 − 0.743i)43-s + (0.743 + 0.669i)47-s + (−0.913 + 0.406i)49-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)5-s + (−0.207 − 0.978i)7-s + (0.994 + 0.104i)11-s + (−0.207 + 0.978i)13-s + (−0.587 + 0.809i)17-s + (−0.951 − 0.309i)19-s + (−0.978 − 0.207i)23-s + (−0.978 + 0.207i)25-s + (−0.406 + 0.913i)29-s + (−0.913 + 0.406i)31-s + (−0.951 + 0.309i)35-s + (−0.809 + 0.587i)37-s + (0.669 − 0.743i)43-s + (0.743 + 0.669i)47-s + (−0.913 + 0.406i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $-0.545 + 0.838i$
Analytic conductor: \(6.85451\)
Root analytic conductor: \(6.85451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1476,\ (0:\ ),\ -0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09728996565 + 0.1793585870i\)
\(L(\frac12)\) \(\approx\) \(0.09728996565 + 0.1793585870i\)
\(L(1)\) \(\approx\) \(0.7841066288 - 0.1287618106i\)
\(L(1)\) \(\approx\) \(0.7841066288 - 0.1287618106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (-0.207 - 0.978i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (-0.406 + 0.913i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (0.743 + 0.669i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 - T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (-0.406 + 0.913i)T \)
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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

−20.32230322671065588193352649288, −19.49237656829888012100236546250, −18.97860326959532959497906750563, −18.11923367269538176860429703573, −17.64767842922434951670934787588, −16.64624853979950710336088607893, −15.6520162512621888663213828106, −15.164904627479373692994279850381, −14.46989262605718381194838374316, −13.70105136466229082524356005841, −12.68358275616630089292315006532, −11.95130278376525624000260820847, −11.26998031891474664885672368620, −10.4537702809780671661677494967, −9.57712338830626788862486321120, −8.88152063983496498514573697859, −7.887131653622997179170567329987, −7.1020610535094452807231524016, −6.11533689391809391359281844017, −5.73526357610452813561363909721, −4.36617626646596155778272003938, −3.50897385614596681746374695755, −2.60305004935067207179332951700, −1.88957306651487566928132101078, −0.07288623122704085374170466592, 1.35055132610785592253605478857, 1.99164844578392252902808806680, 3.69989366023281108690434624544, 4.14853405160507274004499661643, 4.88759383222429055906046746371, 6.14284498336041399922394235299, 6.802259266211267630464063270959, 7.669545305226780693759573749970, 8.78663727863489982427200462867, 9.10323545769298124042200961362, 10.171405399285144410648059020985, 10.947920716195982322580038542843, 11.8674849906002924196443242478, 12.58227552524535443995928462462, 13.26721568520783778712951818517, 14.12168386934981347978045452653, 14.72981722974788314093662771195, 15.90662720355171276289716716592, 16.45688014054531156467402610387, 17.22982569819723448186910262800, 17.51163956107724608168093128820, 18.94961243577966611471663542712, 19.53377281426381774257261294138, 20.155437614885478540890739166734, 20.72451461418253118263433915026

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