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Properties

Label 2-80-5.4-c3-0-0
Degree $2$
Conductor $80$
Sign $-0.965 - 0.259i$
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  + 6.89i·3-s + (−10.7 − 2.89i)5-s + 12.6i·7-s − 20.5·9-s − 59.1·11-s − 42.2i·13-s + (20 − 74.4i)15-s + 126. i·17-s − 19.1·19-s − 87.5·21-s + 78.3i·23-s + (108. + 62.6i)25-s + 44.1i·27-s + 148.·29-s + 139.·31-s + ⋯
L(s)  = 1  + 1.32i·3-s + (−0.965 − 0.259i)5-s + 0.685i·7-s − 0.762·9-s − 1.62·11-s − 0.900i·13-s + (0.344 − 1.28i)15-s + 1.80i·17-s − 0.231·19-s − 0.910·21-s + 0.709i·23-s + (0.865 + 0.500i)25-s + 0.314i·27-s + 0.950·29-s + 0.806·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.965 - 0.259i$
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: \((2,\ 80,\ (\ :3/2),\ -0.965 - 0.259i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.100525 + 0.762124i\)
\(L(\frac12)\) \(\approx\) \(0.100525 + 0.762124i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.7 + 2.89i)T \)
good3 \( 1 - 6.89iT - 27T^{2} \)
7 \( 1 - 12.6iT - 343T^{2} \)
11 \( 1 + 59.1T + 1.33e3T^{2} \)
13 \( 1 + 42.2iT - 2.19e3T^{2} \)
17 \( 1 - 126. iT - 4.91e3T^{2} \)
19 \( 1 + 19.1T + 6.85e3T^{2} \)
23 \( 1 - 78.3iT - 1.21e4T^{2} \)
29 \( 1 - 148.T + 2.43e4T^{2} \)
31 \( 1 - 139.T + 2.97e4T^{2} \)
37 \( 1 + 66.5iT - 5.06e4T^{2} \)
41 \( 1 + 203.T + 6.89e4T^{2} \)
43 \( 1 - 288. iT - 7.95e4T^{2} \)
47 \( 1 + 360. iT - 1.03e5T^{2} \)
53 \( 1 - 686. iT - 1.48e5T^{2} \)
59 \( 1 + 83.1T + 2.05e5T^{2} \)
61 \( 1 + 208.T + 2.26e5T^{2} \)
67 \( 1 + 192. iT - 3.00e5T^{2} \)
71 \( 1 + 500.T + 3.57e5T^{2} \)
73 \( 1 + 122. iT - 3.89e5T^{2} \)
79 \( 1 + 289.T + 4.93e5T^{2} \)
83 \( 1 + 573. iT - 5.71e5T^{2} \)
89 \( 1 - 565.T + 7.04e5T^{2} \)
97 \( 1 - 643. iT - 9.12e5T^{2} \)
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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

−15.05039349881851024422042397753, −13.14761173250572637682422919201, −12.19145247475663599124636096891, −10.78542590624163208789060094282, −10.21541214587252019342893328122, −8.688003410234071073683043577782, −7.87776854377764400457506383481, −5.65735560745928232421015020108, −4.54568818729749088897292605765, −3.16448488694198354765990803270, 0.46927494006564300260224132921, 2.67907010140260838678315132009, 4.70190385875995423008554402988, 6.75096687759686485093148992146, 7.42798408139642627157583569935, 8.370126726547041795488768742635, 10.25527865668363558814914410982, 11.46911320426340671278095720569, 12.32028534985584654729697258728, 13.41733050393229321136191021734

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