↑

Properties

Label 2-880-55.28-c1-0-17
Degree $2$
Conductor $880$
Sign $0.650 - 0.759i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + (0.130 + 0.822i)3-s + (−1.11 + 1.93i)5-s + (4.16 + 0.659i)7-s + (2.19 − 0.712i)9-s + (0.920 − 3.18i)11-s + (−0.824 − 0.420i)13-s + (−1.73 − 0.667i)15-s + (1.71 − 0.875i)17-s + (4.39 + 3.19i)19-s + 3.50i·21-s + (−1.95 − 1.95i)23-s + (−2.49 − 4.33i)25-s + (2.00 + 3.93i)27-s + (0.810 − 0.588i)29-s + (−0.131 − 0.403i)31-s + ⋯
L(s)  = 1  + (0.0751 + 0.474i)3-s + (−0.500 + 0.865i)5-s + (1.57 + 0.249i)7-s + (0.731 − 0.237i)9-s + (0.277 − 0.960i)11-s + (−0.228 − 0.116i)13-s + (−0.448 − 0.172i)15-s + (0.416 − 0.212i)17-s + (1.00 + 0.732i)19-s + 0.765i·21-s + (−0.408 − 0.408i)23-s + (−0.499 − 0.866i)25-s + (0.385 + 0.757i)27-s + (0.150 − 0.109i)29-s + (−0.0235 − 0.0724i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.650 - 0.759i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73401 + 0.797771i\)
\(L(\frac12)\) \(\approx\) \(1.73401 + 0.797771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.11 - 1.93i)T \)
11 \( 1 + (-0.920 + 3.18i)T \)
good3 \( 1 + (-0.130 - 0.822i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-4.16 - 0.659i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (0.824 + 0.420i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.71 + 0.875i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-4.39 - 3.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.95 + 1.95i)T + 23iT^{2} \)
29 \( 1 + (-0.810 + 0.588i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.131 + 0.403i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.771 - 4.87i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-0.339 + 0.467i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.05 - 5.05i)T - 43iT^{2} \)
47 \( 1 + (-1.17 + 0.186i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (4.12 - 8.09i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.47 + 7.53i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.40 - 2.40i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.05 + 3.05i)T - 67iT^{2} \)
71 \( 1 + (2.65 - 8.17i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.854 - 5.39i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-0.705 - 2.17i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.902 + 1.77i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + (2.96 + 1.50i)T + (57.0 + 78.4i)T^{2} \)
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

−10.31782824207853706947421012434, −9.558170232230351438486337623873, −8.351701608452434810817732225405, −7.87323801067375408567210123791, −6.95231952555682472238204340488, −5.82965453902479883828442503728, −4.82889070383034687254745209984, −3.92051040410929850675499954188, −2.95034012048891889271326922945, −1.39539853167209210813970611832, 1.18137905985592068093516887511, 1.98575930668576341494439452408, 3.90288647685781599293319985039, 4.74685140940919902519540299374, 5.29717388980487210704096582410, 6.94327896200307242007256112300, 7.60599953406645769180440845193, 8.090211254315805011603488078244, 9.108838009037040088087191890663, 9.962162028274754466907206640976

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