Properties

Label 2-80-5.2-c4-0-5
Degree $2$
Conductor $80$
Sign $0.829 + 0.558i$
Analytic cond. $8.26959$
Root an. cond. $2.87569$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8.28 + 8.28i)3-s + (−10.2 − 22.7i)5-s + (3.20 + 3.20i)7-s − 56.4i·9-s + 19.5·11-s + (219. − 219. i)13-s + (274. + 103. i)15-s + (281. + 281. i)17-s − 305. i·19-s − 53.1·21-s + (633. − 633. i)23-s + (−413. + 468. i)25-s + (−203. − 203. i)27-s − 885. i·29-s − 1.38e3·31-s + ⋯
L(s)  = 1  + (−0.920 + 0.920i)3-s + (−0.411 − 0.911i)5-s + (0.0654 + 0.0654i)7-s − 0.696i·9-s + 0.161·11-s + (1.30 − 1.30i)13-s + (1.21 + 0.460i)15-s + (0.973 + 0.973i)17-s − 0.846i·19-s − 0.120·21-s + (1.19 − 1.19i)23-s + (−0.661 + 0.750i)25-s + (−0.279 − 0.279i)27-s − 1.05i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(8.26959\)
Root analytic conductor: \(2.87569\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :2),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.01656 - 0.310603i\)
\(L(\frac12)\) \(\approx\) \(1.01656 - 0.310603i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (10.2 + 22.7i)T \)
good3 \( 1 + (8.28 - 8.28i)T - 81iT^{2} \)
7 \( 1 + (-3.20 - 3.20i)T + 2.40e3iT^{2} \)
11 \( 1 - 19.5T + 1.46e4T^{2} \)
13 \( 1 + (-219. + 219. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-281. - 281. i)T + 8.35e4iT^{2} \)
19 \( 1 + 305. iT - 1.30e5T^{2} \)
23 \( 1 + (-633. + 633. i)T - 2.79e5iT^{2} \)
29 \( 1 + 885. iT - 7.07e5T^{2} \)
31 \( 1 + 1.38e3T + 9.23e5T^{2} \)
37 \( 1 + (-453. - 453. i)T + 1.87e6iT^{2} \)
41 \( 1 + 193.T + 2.82e6T^{2} \)
43 \( 1 + (-1.01e3 + 1.01e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.20e3 - 1.20e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (530. - 530. i)T - 7.89e6iT^{2} \)
59 \( 1 - 934. iT - 1.21e7T^{2} \)
61 \( 1 + 1.12e3T + 1.38e7T^{2} \)
67 \( 1 + (4.76e3 + 4.76e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 8.83e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.00e3 + 3.00e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 3.34e3iT - 3.89e7T^{2} \)
83 \( 1 + (-3.16e3 + 3.16e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.73e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.65e3 - 1.65e3i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.27156380018041605529213663067, −12.39992135910893438460990091715, −11.16139124394418453712286023706, −10.48302554546263533703964024021, −9.084025684758939263771169993963, −7.982504911922789007839600245380, −5.98026619304886231120817763354, −5.02609073427527949881214730964, −3.75507096491613724091460507026, −0.70800045586093495469549639117, 1.34169579558453104409008607438, 3.58497255495702210804453373643, 5.64611250937149850616350517205, 6.77856839337462370621871747655, 7.53549208885502233431350141370, 9.273084172393699276186773432406, 10.96617839988728138345822123748, 11.47630048374971107959961190142, 12.46639879102038040115224323033, 13.73617700563566375212993329661

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