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Properties

Label 2-80-5.4-c3-0-6
Degree $2$
Conductor $80$
Sign $-0.626 + 0.779i$
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  − 8.71i·3-s + (7 − 8.71i)5-s + 8.71i·7-s − 49.0·9-s − 20·11-s − 52.3i·13-s + (−76.0 − 61.0i)15-s + 69.7i·17-s + 84·19-s + 76.0·21-s − 61.0i·23-s + (−27.0 − 122. i)25-s + 191. i·27-s + 6·29-s + 224·31-s + ⋯
L(s)  = 1  − 1.67i·3-s + (0.626 − 0.779i)5-s + 0.470i·7-s − 1.81·9-s − 0.548·11-s − 1.11i·13-s + (−1.30 − 1.05i)15-s + 0.995i·17-s + 1.01·19-s + 0.789·21-s − 0.553i·23-s + (−0.216 − 0.976i)25-s + 1.36i·27-s + 0.0384·29-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.626 + 0.779i$
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: \((2,\ 80,\ (\ :3/2),\ -0.626 + 0.779i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.642341 - 1.33955i\)
\(L(\frac12)\) \(\approx\) \(0.642341 - 1.33955i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7 + 8.71i)T \)
good3 \( 1 + 8.71iT - 27T^{2} \)
7 \( 1 - 8.71iT - 343T^{2} \)
11 \( 1 + 20T + 1.33e3T^{2} \)
13 \( 1 + 52.3iT - 2.19e3T^{2} \)
17 \( 1 - 69.7iT - 4.91e3T^{2} \)
19 \( 1 - 84T + 6.85e3T^{2} \)
23 \( 1 + 61.0iT - 1.21e4T^{2} \)
29 \( 1 - 6T + 2.43e4T^{2} \)
31 \( 1 - 224T + 2.97e4T^{2} \)
37 \( 1 - 122. iT - 5.06e4T^{2} \)
41 \( 1 - 266T + 6.89e4T^{2} \)
43 \( 1 - 305. iT - 7.95e4T^{2} \)
47 \( 1 + 374. iT - 1.03e5T^{2} \)
53 \( 1 + 366. iT - 1.48e5T^{2} \)
59 \( 1 - 28T + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 + 427. iT - 3.00e5T^{2} \)
71 \( 1 + 408T + 3.57e5T^{2} \)
73 \( 1 - 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 + 48T + 4.93e5T^{2} \)
83 \( 1 - 200. iT - 5.71e5T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 - 557. iT - 9.12e5T^{2} \)
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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

−13.15344432305783679922466421106, −12.72684468901857924646042956660, −11.76360690257731640353147582372, −10.13607602516701637794646635586, −8.561186822214237742895964016016, −7.83188575322641856916013443564, −6.31848747649302459891060928263, −5.36475860423713532666407253837, −2.53137648893827061986403806244, −0.992653905514558847956914837587, 2.93827029102956006361988112693, 4.37922083359893367045961308496, 5.65750406396750815253673918930, 7.27825649109209767850958297817, 9.167988092240455839130278648581, 9.865824827960262837503444583927, 10.76032741095208503542510690260, 11.65948162730227733586004312172, 13.74491274868954831624691689823, 14.23341567688346556072371859149

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