Properties

Label 2-80-5.2-c4-0-4
Degree $2$
Conductor $80$
Sign $0.734 - 0.678i$
Analytic cond. $8.26959$
Root an. cond. $2.87569$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 − 3i)3-s + (−24.5 + 4.77i)5-s + (42.8 + 42.8i)7-s + 63i·9-s + 181.·11-s + (25.1 − 25.1i)13-s + (−59.3 + 87.9i)15-s + (160. + 160. i)17-s − 80.2i·19-s + 257.·21-s + (−324. + 324. i)23-s + (579. − 234. i)25-s + (432 + 432i)27-s + 1.37e3i·29-s + 151.·31-s + ⋯
L(s)  = 1  + (0.333 − 0.333i)3-s + (−0.981 + 0.190i)5-s + (0.874 + 0.874i)7-s + 0.777i·9-s + 1.50·11-s + (0.148 − 0.148i)13-s + (−0.263 + 0.390i)15-s + (0.555 + 0.555i)17-s − 0.222i·19-s + 0.583·21-s + (−0.613 + 0.613i)23-s + (0.927 − 0.374i)25-s + (0.592 + 0.592i)27-s + 1.63i·29-s + 0.157·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(8.26959\)
Root analytic conductor: \(2.87569\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :2),\ 0.734 - 0.678i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.62948 + 0.637239i\)
\(L(\frac12)\) \(\approx\) \(1.62948 + 0.637239i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (24.5 - 4.77i)T \)
good3 \( 1 + (-3 + 3i)T - 81iT^{2} \)
7 \( 1 + (-42.8 - 42.8i)T + 2.40e3iT^{2} \)
11 \( 1 - 181.T + 1.46e4T^{2} \)
13 \( 1 + (-25.1 + 25.1i)T - 2.85e4iT^{2} \)
17 \( 1 + (-160. - 160. i)T + 8.35e4iT^{2} \)
19 \( 1 + 80.2iT - 1.30e5T^{2} \)
23 \( 1 + (324. - 324. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.37e3iT - 7.07e5T^{2} \)
31 \( 1 - 151.T + 9.23e5T^{2} \)
37 \( 1 + (1.42e3 + 1.42e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 3.01e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.61e3 + 1.61e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-397. - 397. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-923. + 923. i)T - 7.89e6iT^{2} \)
59 \( 1 + 4.54e3iT - 1.21e7T^{2} \)
61 \( 1 - 631.T + 1.38e7T^{2} \)
67 \( 1 + (-2.68e3 - 2.68e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.54e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.85e3 + 5.85e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 9.42e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.39e3 + 8.39e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 8.36e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.01e3 - 1.01e3i)T + 8.85e7iT^{2} \)
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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

−14.04641692902637967055281629177, −12.45043293812703551586174846882, −11.72527479903029969425423536190, −10.71558274872261677652005623854, −8.897343302640568793974228434222, −8.138668419773861119786848142023, −6.97273349879301648832284301032, −5.22918003634421916880119526620, −3.63580855987526528904364976965, −1.72161261321637798411229703679, 0.984474755301154319476913117676, 3.66639054295017102104135676886, 4.47138980331205194674989817191, 6.59937946527691138977395217655, 7.86123475422801116376703130805, 8.899682096163265454964621209463, 10.15125499941407953832317813403, 11.60443041670645756630260119372, 12.04764236902844406090337699693, 13.83914339304766040391985237872

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