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Properties

Label 1-320-320.69-r0-0-0
Degree $1$
Conductor $320$
Sign $0.290 + 0.956i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
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  + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s − 31-s + 33-s + (0.923 + 0.382i)37-s + (0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s − 31-s + 33-s + (0.923 + 0.382i)37-s + (0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6066909597 + 0.4499526817i\)
\(L(\frac12)\) \(\approx\) \(0.6066909597 + 0.4499526817i\)
\(L(1)\) \(\approx\) \(0.8131442875 + 0.05067749648i\)
\(L(1)\) \(\approx\) \(0.8131442875 + 0.05067749648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (-0.382 - 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−24.899418746658845640883183013113, −23.88679238161779432482442407522, −23.24278432381233013211798432420, −22.06289290337195490067884967588, −21.57108981083499694942034104831, −20.47317700202222224460906928239, −19.9575695892158900939101503677, −18.50560269684845605764230989915, −17.608064731357248045238559078657, −16.70403621510616014039814361479, −16.10705269124421168239134324818, −14.85746426684472324155221661074, −14.290898266576400261643302512656, −13.06987533489977323558496982703, −11.802065769370836085906947495531, −10.92960385523396737001346775818, −10.30875942964822665992736726424, −9.16301583485101513235548545658, −8.13245260837244481258103023995, −6.96942992759059737097645698462, −5.60231083911278428370439855526, −4.80320519295172990679493855320, −3.803434664246533637054582471273, −2.51512440699419367009939225532, −0.48421675255505683074968034727, 1.76127399008664577204778809601, 2.346003005289743202315647722108, 4.29405961620645117056220207099, 5.39990921538705611737973225412, 6.30039767847466213917573730626, 7.54115638125782210026187547759, 8.14098830716488293042622284814, 9.446239670584100205187222922487, 10.67674949610112420450707983879, 11.7161001145655330284677115700, 12.41787841420796679637949820394, 13.19348168346898610116195063403, 14.54231540944857150946232630587, 15.061707927653774317176581666308, 16.50689088277018458440611399183, 17.4859534149983393308258826, 17.9846763196588911189601614572, 19.03761822116584448411846044131, 19.703729792033456123671371453275, 20.944945153283392697826451260383, 21.833627524756114872973097051976, 22.73614301759832718879984597410, 23.82221454086316232192828940744, 24.19573019143733569580059927602, 25.30766661931668363836027609040

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