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Properties

Label 1-504-504.373-r0-0-0
Degree $1$
Conductor $504$
Sign $0.971 + 0.235i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530533725 + 0.1829268295i\)
\(L(\frac12)\) \(\approx\) \(1.530533725 + 0.1829268295i\)
\(L(1)\) \(\approx\) \(1.212940071 + 0.01886633341i\)
\(L(1)\) \(\approx\) \(1.212940071 + 0.01886633341i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−23.50551360957980161604067735712, −22.55976082845671106604458863440, −22.06516095010916925962293501332, −21.205205072623547047142025151990, −20.16464218852301931826495713426, −19.40331289623826599204003037362, −18.1818473600626233776583189154, −18.03109156387315393724925443722, −16.79429159414930617659587960882, −15.86115560010972668006408226599, −15.0254323699327769865175184311, −13.93269217987103858564397005340, −13.5853264188441179037993048925, −12.30498231958077064934723092009, −11.18498728470183713028221834874, −10.66854288620096467053812416797, −9.56387318312422591172485736584, −8.6838562424015228906839190017, −7.54269888812732208709970504143, −6.52240872615979394455971003284, −5.84555995299150935041124484595, −4.60770726734142316291391198884, −3.22932892079586812469672946002, −2.58846858902882757727236252752, −0.9735645209060427768214908214, 1.332574918485114284007301314692, 2.11543684696599177342064581069, 3.84849574635211253288580628453, 4.5647840231252002370563805250, 5.771661937594262367979548486702, 6.556043091540493546062112802330, 7.82458357531799607281786915583, 8.77155359421034143142643780570, 9.58246456930958472854202730601, 10.37030632820803964587272568440, 11.79467018456317290529059952269, 12.26194116234346380128097116020, 13.454560053246156267960551640275, 13.97660934285171080762095210242, 15.19951316681612431410358925824, 16.00313669533821622997791162052, 17.0485513334733541162246907264, 17.466545724197126456795541246207, 18.57066499244196258062096144519, 19.56098426696204184157158497808, 20.3632925853618897534770051756, 21.098692489795602276751782751487, 21.87121680629245142137554102261, 22.853915521498119980364066480728, 23.78114110972073986283187403185

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