↑

Properties

Label 2-980-28.27-c1-0-52
Degree $2$
Conductor $980$
Sign $-0.949 + 0.314i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.17 − 0.784i)2-s − 2.99·3-s + (0.767 − 1.84i)4-s + i·5-s + (−3.52 + 2.35i)6-s + (−0.546 − 2.77i)8-s + 5.98·9-s + (0.784 + 1.17i)10-s + 2.23i·11-s + (−2.30 + 5.53i)12-s − 3.17i·13-s − 2.99i·15-s + (−2.82 − 2.83i)16-s − 3.44i·17-s + (7.04 − 4.70i)18-s + 2.05·19-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s − 1.73·3-s + (0.383 − 0.923i)4-s + 0.447i·5-s + (−1.43 + 0.960i)6-s + (−0.193 − 0.981i)8-s + 1.99·9-s + (0.248 + 0.372i)10-s + 0.674i·11-s + (−0.664 + 1.59i)12-s − 0.879i·13-s − 0.774i·15-s + (−0.705 − 0.708i)16-s − 0.835i·17-s + (1.66 − 1.10i)18-s + 0.470·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.949 + 0.314i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132916 - 0.824361i\)
\(L(\frac12)\) \(\approx\) \(0.132916 - 0.824361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.784i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 + 2.99T + 3T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 + 3.17iT - 13T^{2} \)
17 \( 1 + 3.44iT - 17T^{2} \)
19 \( 1 - 2.05T + 19T^{2} \)
23 \( 1 + 2.66iT - 23T^{2} \)
29 \( 1 + 7.38T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 + 9.95iT - 43T^{2} \)
47 \( 1 - 6.12T + 47T^{2} \)
53 \( 1 - 4.65T + 53T^{2} \)
59 \( 1 + 7.11T + 59T^{2} \)
61 \( 1 - 2.53iT - 61T^{2} \)
67 \( 1 - 0.0527iT - 67T^{2} \)
71 \( 1 + 0.212iT - 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 0.461iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 7.02iT - 89T^{2} \)
97 \( 1 - 0.185iT - 97T^{2} \)
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

−10.20276415212209195287531093926, −9.214157383662037860093830639089, −7.28682967920298782733457027361, −6.98418379003159912063262678272, −5.75946118034173763067453244437, −5.40448221682091403662557304794, −4.49143690051416194271405350471, −3.38554446507200618368307105033, −1.90109509951191267894198854107, −0.36191551942283257256459630992, 1.61950641206102807297606897023, 3.62267319303442031053311196107, 4.46300159390410103788749440403, 5.45647833632790441925321758347, 5.80301194002328497904525810061, 6.74008121142595906116120336398, 7.44492678665017593968998602868, 8.580816160976282103983759841109, 9.594332140048011157833369302602, 10.85230036360661118734562108348

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