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Properties

Label 2-80-80.69-c1-0-7
Degree $2$
Conductor $80$
Sign $0.0204 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.490 + 1.32i)2-s + (−1.99 − 1.99i)3-s + (−1.51 − 1.30i)4-s + (−0.569 − 2.16i)5-s + (3.61 − 1.66i)6-s − 1.09·7-s + (2.46 − 1.37i)8-s + 4.93i·9-s + (3.14 + 0.303i)10-s + (−2.33 − 2.33i)11-s + (0.437 + 5.61i)12-s + (1.80 + 1.80i)13-s + (0.534 − 1.44i)14-s + (−3.17 + 5.44i)15-s + (0.619 + 3.95i)16-s − 4.93i·17-s + ⋯
L(s)  = 1  + (−0.346 + 0.938i)2-s + (−1.14 − 1.14i)3-s + (−0.759 − 0.650i)4-s + (−0.254 − 0.966i)5-s + (1.47 − 0.680i)6-s − 0.412·7-s + (0.873 − 0.487i)8-s + 1.64i·9-s + (0.995 + 0.0960i)10-s + (−0.703 − 0.703i)11-s + (0.126 + 1.62i)12-s + (0.501 + 0.501i)13-s + (0.142 − 0.386i)14-s + (−0.818 + 1.40i)15-s + (0.154 + 0.987i)16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0204 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0204 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.0204 + 0.999i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.0204 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298362 - 0.292318i\)
\(L(\frac12)\) \(\approx\) \(0.298362 - 0.292318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.490 - 1.32i)T \)
5 \( 1 + (0.569 + 2.16i)T \)
good3 \( 1 + (1.99 + 1.99i)T + 3iT^{2} \)
7 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 + (2.33 + 2.33i)T + 11iT^{2} \)
13 \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \)
17 \( 1 + 4.93iT - 17T^{2} \)
19 \( 1 + (-2.03 + 2.03i)T - 19iT^{2} \)
23 \( 1 - 1.45T + 23T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - 29iT^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (2.22 - 2.22i)T - 43iT^{2} \)
47 \( 1 - 2.09iT - 47T^{2} \)
53 \( 1 + (-0.215 + 0.215i)T - 53iT^{2} \)
59 \( 1 + (1.16 + 1.16i)T + 59iT^{2} \)
61 \( 1 + (3.46 - 3.46i)T - 61iT^{2} \)
67 \( 1 + (5.04 + 5.04i)T + 67iT^{2} \)
71 \( 1 - 6.40iT - 71T^{2} \)
73 \( 1 - 5.24T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + (-5.67 - 5.67i)T + 83iT^{2} \)
89 \( 1 - 6.87iT - 89T^{2} \)
97 \( 1 - 3.77iT - 97T^{2} \)
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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

−13.74445383095815217640429390637, −13.23873081865197983507490351824, −12.09470320145691596844816799830, −11.01468735086485716604081740682, −9.365883201736405753465165089176, −8.116180767303082554410668415966, −7.04794486881658863967493100647, −5.92621596359461992355779902866, −4.92463200296095668511785108703, −0.71469653344265967894585319826, 3.27636019614546309662413379690, 4.60585392816244410577943273729, 6.18614992892834738049401315899, 8.014549456668907080976949703235, 9.866457904466070272067792929359, 10.28561121868058488186027492001, 11.15185632488293192444019081780, 12.06442192111480164297626198764, 13.26528301044382874480228062306, 14.92367024161945698549383978436

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