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Properties

Label 2-80-80.59-c2-0-4
Degree $2$
Conductor $80$
Sign $-0.467 - 0.883i$
Analytic cond. $2.17984$
Root an. cond. $1.47642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.860 + 1.80i)2-s + (1.09 − 1.09i)3-s + (−2.52 − 3.10i)4-s + (−2.32 + 4.42i)5-s + (1.03 + 2.92i)6-s + 12.9i·7-s + (7.77 − 1.87i)8-s + 6.59i·9-s + (−5.98 − 8.00i)10-s + (4.18 − 4.18i)11-s + (−6.17 − 0.642i)12-s + (−3.42 − 3.42i)13-s + (−23.3 − 11.1i)14-s + (2.30 + 7.40i)15-s + (−3.29 + 15.6i)16-s − 22.5i·17-s + ⋯
L(s)  = 1  + (−0.430 + 0.902i)2-s + (0.365 − 0.365i)3-s + (−0.630 − 0.776i)4-s + (−0.465 + 0.885i)5-s + (0.172 + 0.487i)6-s + 1.84i·7-s + (0.972 − 0.234i)8-s + 0.732i·9-s + (−0.598 − 0.800i)10-s + (0.380 − 0.380i)11-s + (−0.514 − 0.0535i)12-s + (−0.263 − 0.263i)13-s + (−1.66 − 0.795i)14-s + (0.153 + 0.493i)15-s + (−0.205 + 0.978i)16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.467 - 0.883i$
Analytic conductor: \(2.17984\)
Root analytic conductor: \(1.47642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1),\ -0.467 - 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.498385 + 0.827465i\)
\(L(\frac12)\) \(\approx\) \(0.498385 + 0.827465i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.860 - 1.80i)T \)
5 \( 1 + (2.32 - 4.42i)T \)
good3 \( 1 + (-1.09 + 1.09i)T - 9iT^{2} \)
7 \( 1 - 12.9iT - 49T^{2} \)
11 \( 1 + (-4.18 + 4.18i)T - 121iT^{2} \)
13 \( 1 + (3.42 + 3.42i)T + 169iT^{2} \)
17 \( 1 + 22.5iT - 289T^{2} \)
19 \( 1 + (-8.96 - 8.96i)T + 361iT^{2} \)
23 \( 1 + 4.85iT - 529T^{2} \)
29 \( 1 + (-29.3 + 29.3i)T - 841iT^{2} \)
31 \( 1 - 43.7iT - 961T^{2} \)
37 \( 1 + (-20.1 + 20.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 18.6iT - 1.68e3T^{2} \)
43 \( 1 + (-14.4 - 14.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 34.6T + 2.20e3T^{2} \)
53 \( 1 + (36.7 - 36.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (-44.2 + 44.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (-38.7 + 38.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-64.1 + 64.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 29.8T + 5.04e3T^{2} \)
73 \( 1 - 12.9T + 5.32e3T^{2} \)
79 \( 1 - 59.1iT - 6.24e3T^{2} \)
83 \( 1 + (15.4 - 15.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 43.3iT - 7.92e3T^{2} \)
97 \( 1 - 101. iT - 9.40e3T^{2} \)
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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

−14.53288462823024033944849423026, −13.92099028535527018741583782404, −12.34706116181938711538869514278, −11.19280645517663887053003174196, −9.728162320583570469713370544774, −8.571980077448608142629260536296, −7.69387337694124421756352566229, −6.42631339524494017497229943768, −5.16436132586887004285462769941, −2.64704017143102380100929021850, 0.990927428815694840588182472322, 3.74405619761426621955802770349, 4.40839283374392113074984112112, 7.15761604116163509659145310884, 8.341595188916586043918171626057, 9.470326495622403757424717930493, 10.33917565993493561765078003062, 11.54664863674233018391261126686, 12.63066732856990750479736370423, 13.54126041080486221011215658927

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