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Properties

Label 2-80-80.27-c1-0-8
Degree $2$
Conductor $80$
Sign $0.636 + 0.771i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.828 − 1.14i)2-s + 0.692·3-s + (−0.627 − 1.89i)4-s + (−0.245 + 2.22i)5-s + (0.573 − 0.794i)6-s + (−0.343 − 0.343i)7-s + (−2.69 − 0.853i)8-s − 2.52·9-s + (2.34 + 2.12i)10-s + (0.843 − 0.843i)11-s + (−0.434 − 1.31i)12-s + 3.68i·13-s + (−0.678 + 0.109i)14-s + (−0.169 + 1.53i)15-s + (−3.21 + 2.38i)16-s + (0.412 + 0.412i)17-s + ⋯
L(s)  = 1  + (0.585 − 0.810i)2-s + 0.399·3-s + (−0.313 − 0.949i)4-s + (−0.109 + 0.993i)5-s + (0.234 − 0.324i)6-s + (−0.129 − 0.129i)7-s + (−0.953 − 0.301i)8-s − 0.840·9-s + (0.741 + 0.671i)10-s + (0.254 − 0.254i)11-s + (−0.125 − 0.379i)12-s + 1.02i·13-s + (−0.181 + 0.0292i)14-s + (−0.0438 + 0.397i)15-s + (−0.802 + 0.596i)16-s + (0.0999 + 0.0999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12642 - 0.531088i\)
\(L(\frac12)\) \(\approx\) \(1.12642 - 0.531088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.828 + 1.14i)T \)
5 \( 1 + (0.245 - 2.22i)T \)
good3 \( 1 - 0.692T + 3T^{2} \)
7 \( 1 + (0.343 + 0.343i)T + 7iT^{2} \)
11 \( 1 + (-0.843 + 0.843i)T - 11iT^{2} \)
13 \( 1 - 3.68iT - 13T^{2} \)
17 \( 1 + (-0.412 - 0.412i)T + 17iT^{2} \)
19 \( 1 + (-5.37 + 5.37i)T - 19iT^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 23iT^{2} \)
29 \( 1 + (-4.22 - 4.22i)T + 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 + 2.54iT - 41T^{2} \)
43 \( 1 + 4.30iT - 43T^{2} \)
47 \( 1 + (4.56 - 4.56i)T - 47iT^{2} \)
53 \( 1 - 6.07T + 53T^{2} \)
59 \( 1 + (-7.33 - 7.33i)T + 59iT^{2} \)
61 \( 1 + (4.81 - 4.81i)T - 61iT^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 + 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \)
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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

−14.08079406694190481363832209785, −13.47292556271921233357855621167, −11.73326463381085822762382503533, −11.31551292775281328764642806538, −9.963135954222803074020498021366, −8.913091231076844986459612065712, −7.09516803332818603083463163024, −5.72701751667994759984488845038, −3.84870330883114885208135802640, −2.59868002601957994118294373455, 3.31095402654251691635767114458, 4.96114928518326551801356461087, 6.06126074730888645115021594826, 7.86568137367942978447336439731, 8.500561779656269827049156071074, 9.778974757077144884671762554900, 11.83695301605717708495139802737, 12.51742016567534863662541002998, 13.68535160251445585980766102728, 14.46147509618014767302024170241

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