↑

Properties

Label 2-80-80.29-c5-0-20
Degree $2$
Conductor $80$
Sign $0.645 + 0.763i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 − 0.203i)2-s + (−12.8 + 12.8i)3-s + (31.9 + 2.30i)4-s + (−50.2 + 24.4i)5-s + (75.3 − 70.1i)6-s − 156.·7-s + (−179. − 19.5i)8-s − 88.2i·9-s + (289. − 128. i)10-s + (−224. + 224. i)11-s + (−440. + 381. i)12-s + (−554. + 554. i)13-s + (882. + 31.7i)14-s + (331. − 961. i)15-s + (1.01e3 + 147. i)16-s + 587. i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0360i)2-s + (−0.825 + 0.825i)3-s + (0.997 + 0.0719i)4-s + (−0.898 + 0.437i)5-s + (0.854 − 0.795i)6-s − 1.20·7-s + (−0.994 − 0.107i)8-s − 0.363i·9-s + (0.914 − 0.405i)10-s + (−0.559 + 0.559i)11-s + (−0.882 + 0.763i)12-s + (−0.910 + 0.910i)13-s + (1.20 + 0.0433i)14-s + (0.380 − 1.10i)15-s + (0.989 + 0.143i)16-s + 0.492i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.645 + 0.763i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0414098 - 0.0192122i\)
\(L(\frac12)\) \(\approx\) \(0.0414098 - 0.0192122i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.65 + 0.203i)T \)
5 \( 1 + (50.2 - 24.4i)T \)
good3 \( 1 + (12.8 - 12.8i)T - 243iT^{2} \)
7 \( 1 + 156.T + 1.68e4T^{2} \)
11 \( 1 + (224. - 224. i)T - 1.61e5iT^{2} \)
13 \( 1 + (554. - 554. i)T - 3.71e5iT^{2} \)
17 \( 1 - 587. iT - 1.41e6T^{2} \)
19 \( 1 + (1.23e3 + 1.23e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 4.64e3T + 6.43e6T^{2} \)
29 \( 1 + (-1.32e3 - 1.32e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 948.T + 2.86e7T^{2} \)
37 \( 1 + (-1.10e3 - 1.10e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.66e4iT - 1.15e8T^{2} \)
43 \( 1 + (576. + 576. i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.39e4iT - 2.29e8T^{2} \)
53 \( 1 + (-2.12e4 - 2.12e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.72e4 - 2.72e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (2.14e4 + 2.14e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (1.31e4 - 1.31e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.36e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.42e4T + 2.07e9T^{2} \)
79 \( 1 + 7.44e4T + 3.07e9T^{2} \)
83 \( 1 + (2.18e4 - 2.18e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 4.17e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.21e5iT - 8.58e9T^{2} \)
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

−12.83553624977852193319986769362, −11.77364172043441315853467175952, −10.81732974877391872157412890094, −10.08067068748430514641496456123, −9.002363493590440136661739467703, −7.35963437556461956032251285948, −6.46408431971126706971651328187, −4.61128719347419272086600573574, −2.86152258401391403147285142893, −0.04818284759522025494115032515, 0.70911174869757236570487581861, 3.04165658828161849848676940746, 5.56219306064597438232087362447, 6.81061080800060625497715253234, 7.67664788069617331510443618674, 8.954511765623401890858663705961, 10.30015870916060740375968815227, 11.37882560181759812486438204920, 12.45541256892469303662357145083, 12.88973263409401602779123958174

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