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Properties

Label 2-80-16.11-c10-0-33
Degree $2$
Conductor $80$
Sign $-0.131 - 0.991i$
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  + (−21.1 − 24.0i)2-s + (284. + 284. i)3-s + (−133. + 1.01e3i)4-s + (−988. − 988. i)5-s + (838. − 1.28e4i)6-s + 1.47e4·7-s + (2.72e4 − 1.82e4i)8-s + 1.02e5i·9-s + (−2.91e3 + 4.46e4i)10-s + (−8.34e4 + 8.34e4i)11-s + (−3.26e5 + 2.50e5i)12-s + (−1.11e5 + 1.11e5i)13-s + (−3.12e5 − 3.55e5i)14-s − 5.61e5i·15-s + (−1.01e6 − 2.70e5i)16-s + 1.04e6·17-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)2-s + (1.16 + 1.16i)3-s + (−0.130 + 0.991i)4-s + (−0.316 − 0.316i)5-s + (0.107 − 1.65i)6-s + 0.880·7-s + (0.831 − 0.556i)8-s + 1.73i·9-s + (−0.0291 + 0.446i)10-s + (−0.517 + 0.517i)11-s + (−1.31 + 1.00i)12-s + (−0.299 + 0.299i)13-s + (−0.580 − 0.661i)14-s − 0.739i·15-s + (−0.966 − 0.257i)16-s + 0.736·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.131 - 0.991i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.131 - 0.991i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.28162 + 1.46292i\)
\(L(\frac12)\) \(\approx\) \(1.28162 + 1.46292i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (21.1 + 24.0i)T \)
5 \( 1 + (988. + 988. i)T \)
good3 \( 1 + (-284. - 284. i)T + 5.90e4iT^{2} \)
7 \( 1 - 1.47e4T + 2.82e8T^{2} \)
11 \( 1 + (8.34e4 - 8.34e4i)T - 2.59e10iT^{2} \)
13 \( 1 + (1.11e5 - 1.11e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 1.04e6T + 2.01e12T^{2} \)
19 \( 1 + (-2.78e6 - 2.78e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 4.95e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.23e7 + 1.23e7i)T - 4.20e14iT^{2} \)
31 \( 1 + 1.15e7iT - 8.19e14T^{2} \)
37 \( 1 + (-2.16e7 - 2.16e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.17e7iT - 1.34e16T^{2} \)
43 \( 1 + (1.06e8 - 1.06e8i)T - 2.16e16iT^{2} \)
47 \( 1 - 3.26e8iT - 5.25e16T^{2} \)
53 \( 1 + (5.66e8 + 5.66e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (6.68e8 - 6.68e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (1.92e8 - 1.92e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-1.15e9 - 1.15e9i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.29e9T + 3.25e18T^{2} \)
73 \( 1 - 1.34e9iT - 4.29e18T^{2} \)
79 \( 1 - 1.72e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.94e9 - 2.94e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 2.54e9iT - 3.11e19T^{2} \)
97 \( 1 + 2.63e9T + 7.37e19T^{2} \)
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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

−12.37025391648761503716473203780, −11.31312602410235487580837077331, −10.01388712832880925029824439055, −9.574666757136972502021528305611, −8.121163239047231621524792176981, −7.87162461928351900968335708531, −4.89247117674491350646010932864, −3.95054937623139104263558531766, −2.78900222442110492084121979237, −1.49795198118662107722836347936, 0.55799955495128290848816794799, 1.71722328921910010619929704501, 3.03924449778583139976219256112, 5.17678963465885333173021783601, 6.74996011865417751828790213347, 7.75013689427604149725466424051, 8.167533279947349911972145821306, 9.336650241536169684555448439915, 10.78323135394667156695882303395, 12.10986849097535146442505382727

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