Properties

Label 2-80-5.3-c4-0-1
Degree $2$
Conductor $80$
Sign $-0.774 - 0.632i$
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  + (10.2 + 10.2i)3-s + (−24.7 − 3.26i)5-s + (−50.7 + 50.7i)7-s + 129. i·9-s − 2.62·11-s + (−43.4 − 43.4i)13-s + (−220. − 287. i)15-s + (131. − 131. i)17-s + 403. i·19-s − 1.04e3·21-s + (334. + 334. i)23-s + (603. + 161. i)25-s + (−498. + 498. i)27-s + 1.17e3i·29-s − 955.·31-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)3-s + (−0.991 − 0.130i)5-s + (−1.03 + 1.03i)7-s + 1.60i·9-s − 0.0216·11-s + (−0.256 − 0.256i)13-s + (−0.981 − 1.27i)15-s + (0.454 − 0.454i)17-s + 1.11i·19-s − 2.36·21-s + (0.632 + 0.632i)23-s + (0.965 + 0.258i)25-s + (−0.684 + 0.684i)27-s + 1.39i·29-s − 0.994·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.774 - 0.632i$
Analytic conductor: \(8.26959\)
Root analytic conductor: \(2.87569\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :2),\ -0.774 - 0.632i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.492509 + 1.38255i\)
\(L(\frac12)\) \(\approx\) \(0.492509 + 1.38255i\)
\(L(3)\) 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 \( 1 + (24.7 + 3.26i)T \)
good3 \( 1 + (-10.2 - 10.2i)T + 81iT^{2} \)
7 \( 1 + (50.7 - 50.7i)T - 2.40e3iT^{2} \)
11 \( 1 + 2.62T + 1.46e4T^{2} \)
13 \( 1 + (43.4 + 43.4i)T + 2.85e4iT^{2} \)
17 \( 1 + (-131. + 131. i)T - 8.35e4iT^{2} \)
19 \( 1 - 403. iT - 1.30e5T^{2} \)
23 \( 1 + (-334. - 334. i)T + 2.79e5iT^{2} \)
29 \( 1 - 1.17e3iT - 7.07e5T^{2} \)
31 \( 1 + 955.T + 9.23e5T^{2} \)
37 \( 1 + (-673. + 673. i)T - 1.87e6iT^{2} \)
41 \( 1 - 818.T + 2.82e6T^{2} \)
43 \( 1 + (2.48 + 2.48i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.56e3 + 1.56e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-277. - 277. i)T + 7.89e6iT^{2} \)
59 \( 1 - 6.33e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.51e3T + 1.38e7T^{2} \)
67 \( 1 + (-713. + 713. i)T - 2.01e7iT^{2} \)
71 \( 1 + 288.T + 2.54e7T^{2} \)
73 \( 1 + (5.56e3 + 5.56e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 4.06e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.28e3 + 1.28e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 4.41e3iT - 6.27e7T^{2} \)
97 \( 1 + (8.80e3 - 8.80e3i)T - 8.85e7iT^{2} \)
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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

−14.50814943164596421589955344853, −12.96534394437685714212958297778, −11.98556252256676359335843075675, −10.50569750104873128229996480515, −9.381054733941201214344835802250, −8.711115276497580972522856735762, −7.45066712306243630914097364699, −5.39379149772941626320948907157, −3.79438483031314217530481161279, −2.91178233211997921264726114521, 0.64633490696082808369910821592, 2.79518618279319741534954751038, 4.00390758201125539229914274681, 6.70463182443263164554311533739, 7.36224125129933241470438290158, 8.381170199996124934234582324191, 9.650694679541261989283628412060, 11.15080572894527113101945164722, 12.56231630451814291036210813122, 13.15615241401717693993453785579

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