↑

Properties

Label 1-468-468.227-r1-0-0
Degree $1$
Conductor $468$
Sign $0.211 + 0.977i$
Analytic cond. $50.2935$
Root an. cond. $50.2935$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)5-s − i·7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + i·41-s + 43-s + (0.866 − 0.5i)47-s − 49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s − i·7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + i·41-s + 43-s + (0.866 − 0.5i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(50.2935\)
Root analytic conductor: \(50.2935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 468,\ (1:\ ),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2929825354 + 0.2363547415i\)
\(L(\frac12)\) \(\approx\) \(0.2929825354 + 0.2363547415i\)
\(L(1)\) \(\approx\) \(0.6994788896 - 0.1513912187i\)
\(L(1)\) \(\approx\) \(0.6994788896 - 0.1513912187i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−23.54719213041696961508452079395, −22.54669397528897462196634387307, −21.95970931750087246246570476095, −20.88087775876224552795721805042, −20.04531274498900524537775987392, −19.08825152402191847985822752045, −18.36677480438179676394022776358, −17.77432573476369437363477424274, −16.243343470474939229646342570049, −15.710684608142110083811336902734, −14.93143629593535297098416071175, −14.08759306380378122075629372092, −12.70749654461314935552959942254, −12.1702404263044041391718900534, −11.15736694415694898207667387491, −10.348522526236315452345460440803, −9.15773486696188261906898925177, −8.21628044520902078045806164443, −7.38152856455551523665946893971, −6.33630787006090550493385342638, −5.20810711197809716141436415987, −4.189337421817928781049094065094, −2.944403255350500374698814100428, −2.09565838853704049392496272991, −0.12812383889295047958274292372, 0.82542882377467847316553538294, 2.3939454993634432759816829854, 3.87054810916464215169882834316, 4.358292185174810118058284119218, 5.666990388917680853279852876376, 6.842089542057166659769662104464, 7.918487440122337826529977315866, 8.39735195391957522060209193418, 9.73122080036934878271981282205, 10.80620426320886057604838464831, 11.3494074249160965128436185707, 12.702305750375341840971002169580, 13.18190374588371804465413460811, 14.28956757910717623495549225419, 15.35318117074167337743673967314, 16.081738131069083811858340938427, 16.88803481577961182527270442835, 17.72634568697537774912312006939, 18.907138859512165759570298292274, 19.662433028161827314410867716714, 20.31533466644802739167015042307, 21.19799914914338251655277681915, 22.143330241362559990711832570747, 23.37155923960503660907531303973, 23.68825587052973608717526630546

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