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Properties

Label 2-80-16.5-c5-0-33
Degree $2$
Conductor $80$
Sign $-0.534 - 0.845i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 − 5.12i)2-s + (2.82 − 2.82i)3-s + (−20.6 + 24.4i)4-s + (17.6 + 17.6i)5-s + (−21.2 − 7.75i)6-s − 197. i·7-s + (174. + 47.3i)8-s + 227. i·9-s + (48.4 − 132. i)10-s + (−541. − 541. i)11-s + (10.8 + 127. i)12-s + (−657. + 657. i)13-s + (−1.01e3 + 470. i)14-s + 99.9·15-s + (−173. − 1.00e3i)16-s + 152.·17-s + ⋯
L(s)  = 1  + (−0.421 − 0.906i)2-s + (0.181 − 0.181i)3-s + (−0.644 + 0.764i)4-s + (0.316 + 0.316i)5-s + (−0.240 − 0.0879i)6-s − 1.52i·7-s + (0.965 + 0.261i)8-s + 0.934i·9-s + (0.153 − 0.420i)10-s + (−1.34 − 1.34i)11-s + (0.0218 + 0.255i)12-s + (−1.07 + 1.07i)13-s + (−1.38 + 0.642i)14-s + 0.114·15-s + (−0.169 − 0.985i)16-s + 0.128·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.534 - 0.845i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.534 - 0.845i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0611142 + 0.110884i\)
\(L(\frac12)\) \(\approx\) \(0.0611142 + 0.110884i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.38 + 5.12i)T \)
5 \( 1 + (-17.6 - 17.6i)T \)
good3 \( 1 + (-2.82 + 2.82i)T - 243iT^{2} \)
7 \( 1 + 197. iT - 1.68e4T^{2} \)
11 \( 1 + (541. + 541. i)T + 1.61e5iT^{2} \)
13 \( 1 + (657. - 657. i)T - 3.71e5iT^{2} \)
17 \( 1 - 152.T + 1.41e6T^{2} \)
19 \( 1 + (428. - 428. i)T - 2.47e6iT^{2} \)
23 \( 1 - 2.10e3iT - 6.43e6T^{2} \)
29 \( 1 + (1.31e3 - 1.31e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 8.01e3T + 2.86e7T^{2} \)
37 \( 1 + (4.17e3 + 4.17e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 6.83e3iT - 1.15e8T^{2} \)
43 \( 1 + (-367. - 367. i)T + 1.47e8iT^{2} \)
47 \( 1 + 6.12e3T + 2.29e8T^{2} \)
53 \( 1 + (8.83e3 + 8.83e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.36e4 + 2.36e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-3.46e4 + 3.46e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (1.28e4 - 1.28e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.67e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.14e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.06e5T + 3.07e9T^{2} \)
83 \( 1 + (4.91e4 - 4.91e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 6.26e4iT - 5.58e9T^{2} \)
97 \( 1 - 2.78e4T + 8.58e9T^{2} \)
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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

−12.84890151056569866199536428653, −11.13066711126671261247274812617, −10.67501244698772692027139912515, −9.590985339823073437710157000008, −8.036913991591155436427614461656, −7.26376688735593441173104424197, −5.02031249325666512797999051341, −3.43254658606789295612169755044, −1.92169721618204715128923653101, −0.05661840421023214554212239763, 2.38726139722750616542940435171, 4.90973632042344310849007491045, 5.74931492496608282590437722119, 7.29682128922848872843952511415, 8.520450591207417718766941178805, 9.482734107901745428696398892569, 10.29438232456724254355447494088, 12.35300228561889492706216356195, 12.90548642998617342374960376231, 14.75622425053499821990839264340

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