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Properties

Label 2-80-5.2-c2-0-1
Degree $2$
Conductor $80$
Sign $0.0898 - 0.995i$
Analytic cond. $2.17984$
Root an. cond. $1.47642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)3-s + (−3 + 4i)5-s + (7 + 7i)7-s + 7i·9-s − 10·11-s + (9 − 9i)13-s + (−1 − 7i)15-s + (1 + i)17-s + 8i·19-s − 14·21-s + (23 − 23i)23-s + (−7 − 24i)25-s + (−16 − 16i)27-s − 8i·29-s + 14·31-s + ⋯
L(s)  = 1  + (−0.333 + 0.333i)3-s + (−0.600 + 0.800i)5-s + (1 + i)7-s + 0.777i·9-s − 0.909·11-s + (0.692 − 0.692i)13-s + (−0.0666 − 0.466i)15-s + (0.0588 + 0.0588i)17-s + 0.421i·19-s − 0.666·21-s + (1 − i)23-s + (−0.280 − 0.959i)25-s + (−0.592 − 0.592i)27-s − 0.275i·29-s + 0.451·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.0898 - 0.995i$
Analytic conductor: \(2.17984\)
Root analytic conductor: \(1.47642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1),\ 0.0898 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.784607 + 0.717042i\)
\(L(\frac12)\) \(\approx\) \(0.784607 + 0.717042i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (3 - 4i)T \)
good3 \( 1 + (1 - i)T - 9iT^{2} \)
7 \( 1 + (-7 - 7i)T + 49iT^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 + (-9 + 9i)T - 169iT^{2} \)
17 \( 1 + (-1 - i)T + 289iT^{2} \)
19 \( 1 - 8iT - 361T^{2} \)
23 \( 1 + (-23 + 23i)T - 529iT^{2} \)
29 \( 1 + 8iT - 841T^{2} \)
31 \( 1 - 14T + 961T^{2} \)
37 \( 1 + (-33 - 33i)T + 1.36e3iT^{2} \)
41 \( 1 + 14T + 1.68e3T^{2} \)
43 \( 1 + (-15 + 15i)T - 1.84e3iT^{2} \)
47 \( 1 + (-39 - 39i)T + 2.20e3iT^{2} \)
53 \( 1 + (7 - 7i)T - 2.80e3iT^{2} \)
59 \( 1 - 56iT - 3.48e3T^{2} \)
61 \( 1 - 42T + 3.72e3T^{2} \)
67 \( 1 + (-7 - 7i)T + 4.48e3iT^{2} \)
71 \( 1 + 98T + 5.04e3T^{2} \)
73 \( 1 + (-49 + 49i)T - 5.32e3iT^{2} \)
79 \( 1 + 96iT - 6.24e3T^{2} \)
83 \( 1 + (-63 + 63i)T - 6.88e3iT^{2} \)
89 \( 1 + 112iT - 7.92e3T^{2} \)
97 \( 1 + (-33 - 33i)T + 9.40e3iT^{2} \)
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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

−14.67535077214901872783188278350, −13.33540187767374254163350378370, −11.98974020683862037235565252480, −11.00550566286039645583661249222, −10.39645971781342152745902948694, −8.477263128820631142131290060685, −7.70398896063801852529497805595, −5.89049525383107242513596017353, −4.69938114670416991454337161149, −2.67840653776458321694065104163, 1.03379330716452026642521066076, 3.96768690193895358555679991101, 5.25112700489574498169748808917, 7.02509083252818418422582478892, 8.039314837497924889158595047933, 9.258201237141827765267687687275, 10.93679536380474073344733669899, 11.61267478054577667802908693450, 12.82899397838247588189098809071, 13.70581060962508907695002622572

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