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Properties

Label 2-80-80.3-c1-0-7
Degree $2$
Conductor $80$
Sign $0.701 + 0.712i$
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  + (1.19 − 0.759i)2-s − 1.39·3-s + (0.846 − 1.81i)4-s + (2.17 − 0.535i)5-s + (−1.66 + 1.05i)6-s + (−2.13 + 2.13i)7-s + (−0.366 − 2.80i)8-s − 1.05·9-s + (2.18 − 2.28i)10-s + (2.17 + 2.17i)11-s + (−1.17 + 2.52i)12-s + 1.54i·13-s + (−0.925 + 4.16i)14-s + (−3.02 + 0.745i)15-s + (−2.56 − 3.06i)16-s + (−3.86 + 3.86i)17-s + ⋯
L(s)  = 1  + (0.843 − 0.536i)2-s − 0.804·3-s + (0.423 − 0.905i)4-s + (0.970 − 0.239i)5-s + (−0.678 + 0.431i)6-s + (−0.806 + 0.806i)7-s + (−0.129 − 0.991i)8-s − 0.353·9-s + (0.690 − 0.723i)10-s + (0.654 + 0.654i)11-s + (−0.340 + 0.728i)12-s + 0.428i·13-s + (−0.247 + 1.11i)14-s + (−0.780 + 0.192i)15-s + (−0.641 − 0.766i)16-s + (−0.937 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11420 - 0.466648i\)
\(L(\frac12)\) \(\approx\) \(1.11420 - 0.466648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.759i)T \)
5 \( 1 + (-2.17 + 0.535i)T \)
good3 \( 1 + 1.39T + 3T^{2} \)
7 \( 1 + (2.13 - 2.13i)T - 7iT^{2} \)
11 \( 1 + (-2.17 - 2.17i)T + 11iT^{2} \)
13 \( 1 - 1.54iT - 13T^{2} \)
17 \( 1 + (3.86 - 3.86i)T - 17iT^{2} \)
19 \( 1 + (-0.0136 - 0.0136i)T + 19iT^{2} \)
23 \( 1 + (3.15 + 3.15i)T + 23iT^{2} \)
29 \( 1 + (-3.33 + 3.33i)T - 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 + 7.24iT - 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + (-3.34 - 3.34i)T + 47iT^{2} \)
53 \( 1 + 7.30T + 53T^{2} \)
59 \( 1 + (3.52 - 3.52i)T - 59iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + 61iT^{2} \)
67 \( 1 + 0.748iT - 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (-0.811 + 0.811i)T - 73iT^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-6.33 + 6.33i)T - 97iT^{2} \)
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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

−14.09839488534551599229865464798, −12.92838222779565633467366668712, −12.25810282320259094963904962990, −11.22657351593398758029829122752, −10.00809273600621581545024318620, −9.073244535387961587336375269419, −6.32734776673178668006969255201, −6.03295846021385348716602200187, −4.48544466313376896212931560133, −2.31173427755191507676879502342, 3.22407160541480759212436266856, 5.09512751218235227888930473354, 6.24556113042840509821568501081, 6.93366952907401415154724324489, 8.847149066992115258971924757319, 10.38625216647679782215687597257, 11.40355214231361299646869914670, 12.56028944311603290643465983667, 13.76399410503305065683477059451, 14.08173438236835268881391626920

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