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Properties

Label 2-80-16.13-c1-0-4
Degree $2$
Conductor $80$
Sign $0.753 + 0.657i$
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  + (0.114 − 1.40i)2-s + (1.42 + 1.42i)3-s + (−1.97 − 0.323i)4-s + (0.707 − 0.707i)5-s + (2.16 − 1.84i)6-s − 0.690i·7-s + (−0.681 + 2.74i)8-s + 1.05i·9-s + (−0.915 − 1.07i)10-s + (−3.06 + 3.06i)11-s + (−2.34 − 3.26i)12-s + (−2.33 − 2.33i)13-s + (−0.973 − 0.0791i)14-s + 2.01·15-s + (3.79 + 1.27i)16-s − 5.28·17-s + ⋯
L(s)  = 1  + (0.0810 − 0.996i)2-s + (0.821 + 0.821i)3-s + (−0.986 − 0.161i)4-s + (0.316 − 0.316i)5-s + (0.885 − 0.752i)6-s − 0.261i·7-s + (−0.241 + 0.970i)8-s + 0.350i·9-s + (−0.289 − 0.340i)10-s + (−0.922 + 0.922i)11-s + (−0.678 − 0.943i)12-s + (−0.648 − 0.648i)13-s + (−0.260 − 0.0211i)14-s + 0.519·15-s + (0.947 + 0.318i)16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04236 - 0.390743i\)
\(L(\frac12)\) \(\approx\) \(1.04236 - 0.390743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 + 1.40i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.42 - 1.42i)T + 3iT^{2} \)
7 \( 1 + 0.690iT - 7T^{2} \)
11 \( 1 + (3.06 - 3.06i)T - 11iT^{2} \)
13 \( 1 + (2.33 + 2.33i)T + 13iT^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + (-5.38 - 5.38i)T + 19iT^{2} \)
23 \( 1 + 1.60iT - 23T^{2} \)
29 \( 1 + (-1.70 - 1.70i)T + 29iT^{2} \)
31 \( 1 + 4.69T + 31T^{2} \)
37 \( 1 + (-7.89 + 7.89i)T - 37iT^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 + (0.256 - 0.256i)T - 43iT^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + (4.99 - 4.99i)T - 53iT^{2} \)
59 \( 1 + (-1.46 + 1.46i)T - 59iT^{2} \)
61 \( 1 + (-9.33 - 9.33i)T + 61iT^{2} \)
67 \( 1 + (1.94 + 1.94i)T + 67iT^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 + 1.29iT - 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (-7.30 - 7.30i)T + 83iT^{2} \)
89 \( 1 - 1.81iT - 89T^{2} \)
97 \( 1 - 5.27T + 97T^{2} \)
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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

−14.22890094039456774819182675157, −13.11467991561381190608720528795, −12.25280011233827527645767404237, −10.65405570675765892881303329849, −9.892896818146779231860846065307, −9.083974796699196057912080254561, −7.78165413865513886131663624735, −5.26688393391931109975047175565, −4.05901604608916736471959729945, −2.52433147495870099152724511179, 2.79467274517295580445009505376, 5.04124931081060413285163400043, 6.56741479125777722843350410967, 7.55123048161366281581913923317, 8.575080363596953921949685562426, 9.597245367975935464203977782933, 11.36328204350741554845725855087, 13.10816596063821793858297646821, 13.49053721839988048102278912219, 14.39272466205385451468999495655

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