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Properties

Label 2-80-80.29-c5-0-31
Degree $2$
Conductor $80$
Sign $-0.709 + 0.704i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.30 − 3.67i)2-s + (−2.38 + 2.38i)3-s + (5.00 + 31.6i)4-s + (4.21 + 55.7i)5-s + (19.0 − 1.49i)6-s − 204.·7-s + (94.6 − 154. i)8-s + 231. i·9-s + (186. − 255. i)10-s + (7.30 − 7.30i)11-s + (−87.4 − 63.5i)12-s + (822. − 822. i)13-s + (878. + 750. i)14-s + (−143. − 123. i)15-s + (−973. + 316. i)16-s − 790. i·17-s + ⋯
L(s)  = 1  + (−0.760 − 0.649i)2-s + (−0.153 + 0.153i)3-s + (0.156 + 0.987i)4-s + (0.0753 + 0.997i)5-s + (0.215 − 0.0169i)6-s − 1.57·7-s + (0.522 − 0.852i)8-s + 0.953i·9-s + (0.590 − 0.807i)10-s + (0.0182 − 0.0182i)11-s + (−0.175 − 0.127i)12-s + (1.34 − 1.34i)13-s + (1.19 + 1.02i)14-s + (−0.164 − 0.141i)15-s + (−0.951 + 0.308i)16-s − 0.663i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.116908 - 0.283604i\)
\(L(\frac12)\) \(\approx\) \(0.116908 - 0.283604i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.30 + 3.67i)T \)
5 \( 1 + (-4.21 - 55.7i)T \)
good3 \( 1 + (2.38 - 2.38i)T - 243iT^{2} \)
7 \( 1 + 204.T + 1.68e4T^{2} \)
11 \( 1 + (-7.30 + 7.30i)T - 1.61e5iT^{2} \)
13 \( 1 + (-822. + 822. i)T - 3.71e5iT^{2} \)
17 \( 1 + 790. iT - 1.41e6T^{2} \)
19 \( 1 + (939. + 939. i)T + 2.47e6iT^{2} \)
23 \( 1 + 2.28e3T + 6.43e6T^{2} \)
29 \( 1 + (886. + 886. i)T + 2.05e7iT^{2} \)
31 \( 1 - 5.11e3T + 2.86e7T^{2} \)
37 \( 1 + (629. + 629. i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.24e4iT - 1.15e8T^{2} \)
43 \( 1 + (9.64e3 + 9.64e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 3.43e3iT - 2.29e8T^{2} \)
53 \( 1 + (-1.76e3 - 1.76e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.01e4 + 2.01e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-1.81e4 - 1.81e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (4.73e4 - 4.73e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 2.21e4iT - 1.80e9T^{2} \)
73 \( 1 + 9.66e3T + 2.07e9T^{2} \)
79 \( 1 + 7.26e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.91e4 + 3.91e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.09e5iT - 5.58e9T^{2} \)
97 \( 1 + 4.90e4iT - 8.58e9T^{2} \)
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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

−13.04924458341314806219192279077, −11.60315807249621990233740155426, −10.47051087635567437917399735193, −10.08625000252479333999682436082, −8.592341110339303393436220779018, −7.27106286252110992687361043899, −6.06986208986230249146682574521, −3.62723738266999554118040492513, −2.57121442051713346384010688429, −0.17975074173818472282715099502, 1.34441434072722944407008997732, 4.00449305275089648169193761729, 6.10888387381481513919235977548, 6.49260473319622273318324136238, 8.359919628739013336000577330516, 9.231601166603442545265668768155, 10.02953391151874135850782865962, 11.65969064550549878797837097627, 12.79820079162948389554875452243, 13.78344993615630883923247648332

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