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Properties

Label 2-80-5.4-c3-0-4
Degree $2$
Conductor $80$
Sign $0.786 + 0.617i$
Analytic cond. $4.72015$
Root an. cond. $2.17259$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.89i·3-s + (8.79 + 6.89i)5-s − 16.6i·7-s + 18.5·9-s + 19.1·11-s − 61.7i·13-s + (20 − 25.5i)15-s − 30.3i·17-s + 59.1·19-s − 48.4·21-s + 205. i·23-s + (29.8 + 121. i)25-s − 132. i·27-s − 8.38·29-s − 331.·31-s + ⋯
L(s)  = 1  − 0.557i·3-s + (0.786 + 0.617i)5-s − 0.901i·7-s + 0.688·9-s + 0.526·11-s − 1.31i·13-s + (0.344 − 0.439i)15-s − 0.433i·17-s + 0.714·19-s − 0.502·21-s + 1.86i·23-s + (0.238 + 0.971i)25-s − 0.942i·27-s − 0.0536·29-s − 1.91·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(4.72015\)
Root analytic conductor: \(2.17259\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :3/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.65966 - 0.573122i\)
\(L(\frac12)\) \(\approx\) \(1.65966 - 0.573122i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.79 - 6.89i)T \)
good3 \( 1 + 2.89iT - 27T^{2} \)
7 \( 1 + 16.6iT - 343T^{2} \)
11 \( 1 - 19.1T + 1.33e3T^{2} \)
13 \( 1 + 61.7iT - 2.19e3T^{2} \)
17 \( 1 + 30.3iT - 4.91e3T^{2} \)
19 \( 1 - 59.1T + 6.85e3T^{2} \)
23 \( 1 - 205. iT - 1.21e4T^{2} \)
29 \( 1 + 8.38T + 2.43e4T^{2} \)
31 \( 1 + 331.T + 2.97e4T^{2} \)
37 \( 1 - 266. iT - 5.06e4T^{2} \)
41 \( 1 + 320.T + 6.89e4T^{2} \)
43 \( 1 - 83.1iT - 7.95e4T^{2} \)
47 \( 1 - 276. iT - 1.03e5T^{2} \)
53 \( 1 + 390. iT - 1.48e5T^{2} \)
59 \( 1 - 779.T + 2.05e5T^{2} \)
61 \( 1 + 483.T + 2.26e5T^{2} \)
67 \( 1 + 123. iT - 3.00e5T^{2} \)
71 \( 1 + 187.T + 3.57e5T^{2} \)
73 \( 1 - 778. iT - 3.89e5T^{2} \)
79 \( 1 + 446.T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3iT - 5.71e5T^{2} \)
89 \( 1 - 94.8T + 7.04e5T^{2} \)
97 \( 1 - 252. iT - 9.12e5T^{2} \)
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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

−13.55722661809914770390332033167, −13.06412050056924149507198662196, −11.56336564294025693295515187884, −10.35692752175668414267061141512, −9.548284382264895237578212558571, −7.64528526049078292612430845377, −6.91318418597326284111298336839, −5.46486368076932560364555209219, −3.43254720933684514653101032187, −1.39220508178131171927500657323, 1.90161720986672114510078184193, 4.18169280658427044106850488815, 5.46130290568458171517876492960, 6.81262746790073769514095299090, 8.806562427020777543442441244926, 9.350880337625508020545042412630, 10.53350589187640339510954123960, 11.98343764398479228228458242322, 12.83083744707581780343632366558, 14.10697122046961965777488945304

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