Properties

Label 2-80-80.37-c10-0-12
Degree $2$
Conductor $80$
Sign $-0.987 - 0.156i$
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  + (2.05 − 31.9i)2-s + 192. i·3-s + (−1.01e3 − 131. i)4-s + (−193. + 3.11e3i)5-s + (6.13e3 + 393. i)6-s + (−1.85e4 + 1.85e4i)7-s + (−6.26e3 + 3.21e4i)8-s + 2.21e4·9-s + (9.92e4 + 1.25e4i)10-s + (1.41e4 + 1.41e4i)11-s + (2.51e4 − 1.95e5i)12-s + 5.58e5i·13-s + (5.54e5 + 6.30e5i)14-s + (−5.98e5 − 3.71e4i)15-s + (1.01e6 + 2.66e5i)16-s + (1.45e5 + 1.45e5i)17-s + ⋯
L(s)  = 1  + (0.0641 − 0.997i)2-s + 0.790i·3-s + (−0.991 − 0.127i)4-s + (−0.0618 + 0.998i)5-s + (0.788 + 0.0506i)6-s + (−1.10 + 1.10i)7-s + (−0.191 + 0.981i)8-s + 0.375·9-s + (0.992 + 0.125i)10-s + (0.0880 + 0.0880i)11-s + (0.101 − 0.783i)12-s + 1.50i·13-s + (1.03 + 1.17i)14-s + (−0.788 − 0.0488i)15-s + (0.967 + 0.253i)16-s + (0.102 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.987 - 0.156i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0805759 + 1.02043i\)
\(L(\frac12)\) \(\approx\) \(0.0805759 + 1.02043i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.05 + 31.9i)T \)
5 \( 1 + (193. - 3.11e3i)T \)
good3 \( 1 - 192. iT - 5.90e4T^{2} \)
7 \( 1 + (1.85e4 - 1.85e4i)T - 2.82e8iT^{2} \)
11 \( 1 + (-1.41e4 - 1.41e4i)T + 2.59e10iT^{2} \)
13 \( 1 - 5.58e5iT - 1.37e11T^{2} \)
17 \( 1 + (-1.45e5 - 1.45e5i)T + 2.01e12iT^{2} \)
19 \( 1 + (1.15e5 + 1.15e5i)T + 6.13e12iT^{2} \)
23 \( 1 + (-7.04e6 - 7.04e6i)T + 4.14e13iT^{2} \)
29 \( 1 + (4.93e6 + 4.93e6i)T + 4.20e14iT^{2} \)
31 \( 1 + 3.10e7T + 8.19e14T^{2} \)
37 \( 1 - 1.55e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.88e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.25e6T + 2.16e16T^{2} \)
47 \( 1 + (-8.45e7 - 8.45e7i)T + 5.25e16iT^{2} \)
53 \( 1 + 3.51e5T + 1.74e17T^{2} \)
59 \( 1 + (1.96e8 - 1.96e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (6.86e8 - 6.86e8i)T - 7.13e17iT^{2} \)
67 \( 1 - 7.27e7T + 1.82e18T^{2} \)
71 \( 1 + 1.31e9iT - 3.25e18T^{2} \)
73 \( 1 + (-1.78e9 - 1.78e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 4.94e9iT - 9.46e18T^{2} \)
83 \( 1 + 6.37e8iT - 1.55e19T^{2} \)
89 \( 1 - 7.78e9T + 3.11e19T^{2} \)
97 \( 1 + (1.05e10 + 1.05e10i)T + 7.37e19iT^{2} \)
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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

−12.62176137762954657246333985502, −11.60187717699355821781251387596, −10.66306075722250668982073522022, −9.517219292939332743396336832263, −9.141956482654149175401145348962, −7.00900805538484699928038201349, −5.59114550683690404892889083569, −4.05245710136129295612058654348, −3.14331174801893660653313056872, −1.92254229605431262300868774522, 0.33294789642380735851166566411, 0.974185013241871287998185012849, 3.47421352499911958995348562844, 4.79737771487174044843355649992, 6.16683069703188769011671840619, 7.20739127953040138020269258404, 8.038215619468174151752770288149, 9.332490281882583895956785472967, 10.40527940809437973517350609132, 12.63680349809714929320316832198

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