Properties

Label 2-80-80.13-c10-0-53
Degree $2$
Conductor $80$
Sign $0.769 - 0.639i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−29.2 + 12.9i)2-s − 135. i·3-s + (690. − 755. i)4-s + (−2.25e3 + 2.16e3i)5-s + (1.75e3 + 3.97e3i)6-s + (7.87e3 + 7.87e3i)7-s + (−1.04e4 + 3.10e4i)8-s + 4.06e4·9-s + (3.81e4 − 9.24e4i)10-s + (2.00e5 − 2.00e5i)11-s + (−1.02e5 − 9.37e4i)12-s + 2.15e5i·13-s + (−3.32e5 − 1.29e5i)14-s + (2.93e5 + 3.06e5i)15-s + (−9.39e4 − 1.04e6i)16-s + (1.31e3 − 1.31e3i)17-s + ⋯
L(s)  = 1  + (−0.915 + 0.403i)2-s − 0.558i·3-s + (0.674 − 0.738i)4-s + (−0.721 + 0.692i)5-s + (0.225 + 0.511i)6-s + (0.468 + 0.468i)7-s + (−0.319 + 0.947i)8-s + 0.687·9-s + (0.381 − 0.924i)10-s + (1.24 − 1.24i)11-s + (−0.412 − 0.376i)12-s + 0.579i·13-s + (−0.618 − 0.239i)14-s + (0.386 + 0.403i)15-s + (−0.0895 − 0.995i)16-s + (0.000928 − 0.000928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.769 - 0.639i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.24475 + 0.449635i\)
\(L(\frac12)\) \(\approx\) \(1.24475 + 0.449635i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (29.2 - 12.9i)T \)
5 \( 1 + (2.25e3 - 2.16e3i)T \)
good3 \( 1 + 135. iT - 5.90e4T^{2} \)
7 \( 1 + (-7.87e3 - 7.87e3i)T + 2.82e8iT^{2} \)
11 \( 1 + (-2.00e5 + 2.00e5i)T - 2.59e10iT^{2} \)
13 \( 1 - 2.15e5iT - 1.37e11T^{2} \)
17 \( 1 + (-1.31e3 + 1.31e3i)T - 2.01e12iT^{2} \)
19 \( 1 + (6.89e5 - 6.89e5i)T - 6.13e12iT^{2} \)
23 \( 1 + (7.89e6 - 7.89e6i)T - 4.14e13iT^{2} \)
29 \( 1 + (-1.04e7 + 1.04e7i)T - 4.20e14iT^{2} \)
31 \( 1 + 1.53e7T + 8.19e14T^{2} \)
37 \( 1 - 2.35e7iT - 4.80e15T^{2} \)
41 \( 1 + 5.93e6iT - 1.34e16T^{2} \)
43 \( 1 - 6.26e7T + 2.16e16T^{2} \)
47 \( 1 + (-4.45e7 + 4.45e7i)T - 5.25e16iT^{2} \)
53 \( 1 - 3.00e8T + 1.74e17T^{2} \)
59 \( 1 + (-4.08e8 - 4.08e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-8.21e8 - 8.21e8i)T + 7.13e17iT^{2} \)
67 \( 1 + 3.81e8T + 1.82e18T^{2} \)
71 \( 1 - 2.91e9iT - 3.25e18T^{2} \)
73 \( 1 + (-1.43e9 + 1.43e9i)T - 4.29e18iT^{2} \)
79 \( 1 + 2.31e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.96e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.82e9T + 3.11e19T^{2} \)
97 \( 1 + (-3.51e9 + 3.51e9i)T - 7.37e19iT^{2} \)
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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

−11.86158371484161700530610283636, −11.51085869352914825480943593299, −10.16166745771325935334281411303, −8.836533774366062627292225094191, −7.88771052300009061194362135494, −6.87234285501574240398290910947, −5.95137778311422863950392240868, −3.87057570681342491020129881547, −2.06644543793166793233214856392, −0.888396284461549326023072394220, 0.66317753392667490500758615980, 1.80274546972024689881627685635, 3.84161966293461757145195082689, 4.55223799389698383469766645588, 6.85186315885563989265627264181, 7.85193908077666128137881074949, 8.986077013319683548664227850485, 9.949819090245765899180918818465, 10.90829223496125070477608618863, 12.12792240005435389725745426436

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