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Properties

Label 2-80-80.19-c2-0-14
Degree $2$
Conductor $80$
Sign $0.994 + 0.106i$
Analytic cond. $2.17984$
Root an. cond. $1.47642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.26i)2-s + (3.99 + 3.99i)3-s + (0.785 − 3.92i)4-s + (−4.97 − 0.520i)5-s + (11.2 + 1.11i)6-s − 4.43i·7-s + (−3.75 − 7.06i)8-s + 22.9i·9-s + (−8.35 + 5.50i)10-s + (1.40 + 1.40i)11-s + (18.8 − 12.5i)12-s + (−5.82 + 5.82i)13-s + (−5.61 − 6.85i)14-s + (−17.7 − 21.9i)15-s + (−14.7 − 6.15i)16-s − 22.9i·17-s + ⋯
L(s)  = 1  + (0.773 − 0.633i)2-s + (1.33 + 1.33i)3-s + (0.196 − 0.980i)4-s + (−0.994 − 0.104i)5-s + (1.87 + 0.185i)6-s − 0.632i·7-s + (−0.469 − 0.882i)8-s + 2.54i·9-s + (−0.835 + 0.550i)10-s + (0.127 + 0.127i)11-s + (1.56 − 1.04i)12-s + (−0.448 + 0.448i)13-s + (−0.401 − 0.489i)14-s + (−1.18 − 1.46i)15-s + (−0.922 − 0.384i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(2.17984\)
Root analytic conductor: \(1.47642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1),\ 0.994 + 0.106i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.25558 - 0.120374i\)
\(L(\frac12)\) \(\approx\) \(2.25558 - 0.120374i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.54 + 1.26i)T \)
5 \( 1 + (4.97 + 0.520i)T \)
good3 \( 1 + (-3.99 - 3.99i)T + 9iT^{2} \)
7 \( 1 + 4.43iT - 49T^{2} \)
11 \( 1 + (-1.40 - 1.40i)T + 121iT^{2} \)
13 \( 1 + (5.82 - 5.82i)T - 169iT^{2} \)
17 \( 1 + 22.9iT - 289T^{2} \)
19 \( 1 + (9.80 - 9.80i)T - 361iT^{2} \)
23 \( 1 + 15.5iT - 529T^{2} \)
29 \( 1 + (-2.41 - 2.41i)T + 841iT^{2} \)
31 \( 1 - 15.4iT - 961T^{2} \)
37 \( 1 + (-39.5 - 39.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 13.4iT - 1.68e3T^{2} \)
43 \( 1 + (-39.6 + 39.6i)T - 1.84e3iT^{2} \)
47 \( 1 + 15.2T + 2.20e3T^{2} \)
53 \( 1 + (4.23 + 4.23i)T + 2.80e3iT^{2} \)
59 \( 1 + (10.4 + 10.4i)T + 3.48e3iT^{2} \)
61 \( 1 + (-65.7 - 65.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (-3.35 - 3.35i)T + 4.48e3iT^{2} \)
71 \( 1 + 69.1T + 5.04e3T^{2} \)
73 \( 1 + 70.7T + 5.32e3T^{2} \)
79 \( 1 + 96.5iT - 6.24e3T^{2} \)
83 \( 1 + (41.8 + 41.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 2.91iT - 7.92e3T^{2} \)
97 \( 1 - 172. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.33192911129451220368369424317, −13.32107295101137279858555090586, −11.87348878721856376064393617552, −10.73623675395044673557715836664, −9.843594761664035678625795963968, −8.748172452185918485207148559040, −7.31358626522350372440055706859, −4.75468671454538524840858542610, −4.06771537660386368541014870749, −2.82872598120300072153987936659, 2.57778855235140001204422209039, 3.88518187631623782446795581833, 6.13717035855444373457810399915, 7.36302015722792847211087184483, 8.090715386211156604773946114852, 8.949612082677969399841326550022, 11.48500168028730562802214579034, 12.59575655432853417121223027159, 12.96627010470408815149322605509, 14.32251668061056785002932841090

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