↑

Properties

Label 2-80-80.27-c1-0-2
Degree $2$
Conductor $80$
Sign $0.999 + 0.0200i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
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  + (−1.38 − 0.307i)2-s + 2.85·3-s + (1.81 + 0.849i)4-s + (−1.71 + 1.43i)5-s + (−3.94 − 0.879i)6-s + (−0.458 − 0.458i)7-s + (−2.23 − 1.73i)8-s + 5.15·9-s + (2.80 − 1.45i)10-s + (−0.492 + 0.492i)11-s + (5.17 + 2.42i)12-s − 4.52i·13-s + (0.492 + 0.774i)14-s + (−4.89 + 4.09i)15-s + (2.55 + 3.07i)16-s + (−3.12 − 3.12i)17-s + ⋯
L(s)  = 1  + (−0.976 − 0.217i)2-s + 1.64·3-s + (0.905 + 0.424i)4-s + (−0.766 + 0.641i)5-s + (−1.60 − 0.358i)6-s + (−0.173 − 0.173i)7-s + (−0.791 − 0.611i)8-s + 1.71·9-s + (0.888 − 0.459i)10-s + (−0.148 + 0.148i)11-s + (1.49 + 0.700i)12-s − 1.25i·13-s + (0.131 + 0.207i)14-s + (−1.26 + 1.05i)15-s + (0.638 + 0.769i)16-s + (−0.758 − 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.999 + 0.0200i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.999 + 0.0200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898087 - 0.00899030i\)
\(L(\frac12)\) \(\approx\) \(0.898087 - 0.00899030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.307i)T \)
5 \( 1 + (1.71 - 1.43i)T \)
good3 \( 1 - 2.85T + 3T^{2} \)
7 \( 1 + (0.458 + 0.458i)T + 7iT^{2} \)
11 \( 1 + (0.492 - 0.492i)T - 11iT^{2} \)
13 \( 1 + 4.52iT - 13T^{2} \)
17 \( 1 + (3.12 + 3.12i)T + 17iT^{2} \)
19 \( 1 + (4.04 - 4.04i)T - 19iT^{2} \)
23 \( 1 + (1.80 - 1.80i)T - 23iT^{2} \)
29 \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \)
31 \( 1 + 0.139iT - 31T^{2} \)
37 \( 1 - 5.84iT - 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \)
53 \( 1 - 2.75T + 53T^{2} \)
59 \( 1 + (-3.62 - 3.62i)T + 59iT^{2} \)
61 \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \)
67 \( 1 - 3.32iT - 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (-2.55 - 2.55i)T + 73iT^{2} \)
79 \( 1 - 3.86T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + (4.95 + 4.95i)T + 97iT^{2} \)
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

−14.70995917268749808985698972704, −13.36529029365862882527797706377, −12.19500905750124686572094158962, −10.70895348002835505541852470220, −9.864848918398146468630431555918, −8.534002622240983379271008313472, −7.901649370793517205816128890697, −6.86453037199916957470210154737, −3.68481460562402221746409382995, −2.58202536610785759150904979431, 2.28443127955614854008954789119, 4.16619993438205904243084955684, 6.71516230036312236492763649200, 7.969638826176142879567816883993, 8.753160246570514032170362482883, 9.342009555296052050996887091755, 10.87355768066164716680751495856, 12.24657568654827227753437136085, 13.49726666599547191693869829338, 14.70163809646811249694947051059

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