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Properties

Label 2-80-80.69-c1-0-8
Degree $2$
Conductor $80$
Sign $-0.293 + 0.956i$
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  + (0.161 − 1.40i)2-s + (−0.734 − 0.734i)3-s + (−1.94 − 0.452i)4-s + (1.90 − 1.17i)5-s + (−1.14 + 0.913i)6-s − 1.71·7-s + (−0.949 + 2.66i)8-s − 1.92i·9-s + (−1.34 − 2.86i)10-s + (2.82 + 2.82i)11-s + (1.09 + 1.76i)12-s + (2.59 + 2.59i)13-s + (−0.276 + 2.40i)14-s + (−2.25 − 0.537i)15-s + (3.59 + 1.76i)16-s + 1.89i·17-s + ⋯
L(s)  = 1  + (0.113 − 0.993i)2-s + (−0.423 − 0.423i)3-s + (−0.974 − 0.226i)4-s + (0.851 − 0.524i)5-s + (−0.469 + 0.372i)6-s − 0.648·7-s + (−0.335 + 0.941i)8-s − 0.640i·9-s + (−0.423 − 0.905i)10-s + (0.852 + 0.852i)11-s + (0.317 + 0.508i)12-s + (0.719 + 0.719i)13-s + (−0.0738 + 0.643i)14-s + (−0.583 − 0.138i)15-s + (0.897 + 0.440i)16-s + 0.460i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.293 + 0.956i$
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: \((2,\ 80,\ (\ :1/2),\ -0.293 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530437 - 0.717422i\)
\(L(\frac12)\) \(\approx\) \(0.530437 - 0.717422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.161 + 1.40i)T \)
5 \( 1 + (-1.90 + 1.17i)T \)
good3 \( 1 + (0.734 + 0.734i)T + 3iT^{2} \)
7 \( 1 + 1.71T + 7T^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
13 \( 1 + (-2.59 - 2.59i)T + 13iT^{2} \)
17 \( 1 - 1.89iT - 17T^{2} \)
19 \( 1 + (-2.89 + 2.89i)T - 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (6.72 - 6.72i)T - 29iT^{2} \)
31 \( 1 + 7.11T + 31T^{2} \)
37 \( 1 + (2.25 - 2.25i)T - 37iT^{2} \)
41 \( 1 + 1.59iT - 41T^{2} \)
43 \( 1 + (-8.06 + 8.06i)T - 43iT^{2} \)
47 \( 1 - 4.43iT - 47T^{2} \)
53 \( 1 + (0.481 - 0.481i)T - 53iT^{2} \)
59 \( 1 + (-3.08 - 3.08i)T + 59iT^{2} \)
61 \( 1 + (-3.46 + 3.46i)T - 61iT^{2} \)
67 \( 1 + (1.80 + 1.80i)T + 67iT^{2} \)
71 \( 1 + 0.379iT - 71T^{2} \)
73 \( 1 + 8.37T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 6.50iT - 97T^{2} \)
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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

−13.71021569203682379967406508084, −12.74284993967291224400575811679, −12.14588760635826544622009928802, −10.92820682039079306552621468435, −9.463310613970549339634390998781, −9.098538138575268777919153573982, −6.77567862482055047861243959565, −5.55646904468619983149520103995, −3.84241418047159126184184669106, −1.59597343247988219945523923530, 3.62287289909574280848490993344, 5.55625292307375161099902779133, 6.17112246008388854500313263856, 7.65801959472136938697122119971, 9.174811255509286124549453427812, 10.08714537957999266014900457315, 11.30805685037572530670164721492, 13.03972804044890383851471878062, 13.77530817310429492927812657564, 14.68996239064655222808487895314

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