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Properties

Label 2-80-16.11-c10-0-29
Degree $2$
Conductor $80$
Sign $0.998 + 0.0539i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−28.0 − 15.4i)2-s + (−226. − 226. i)3-s + (545. + 866. i)4-s + (988. + 988. i)5-s + (2.83e3 + 9.83e3i)6-s + 6.51e3·7-s + (−1.89e3 − 3.27e4i)8-s + 4.32e4i·9-s + (−1.24e4 − 4.29e4i)10-s + (5.06e4 − 5.06e4i)11-s + (7.25e4 − 3.19e5i)12-s + (2.11e5 − 2.11e5i)13-s + (−1.82e5 − 1.00e5i)14-s − 4.46e5i·15-s + (−4.52e5 + 9.45e5i)16-s − 6.37e4·17-s + ⋯
L(s)  = 1  + (−0.875 − 0.483i)2-s + (−0.930 − 0.930i)3-s + (0.532 + 0.846i)4-s + (0.316 + 0.316i)5-s + (0.364 + 1.26i)6-s + 0.387·7-s + (−0.0576 − 0.998i)8-s + 0.731i·9-s + (−0.124 − 0.429i)10-s + (0.314 − 0.314i)11-s + (0.291 − 1.28i)12-s + (0.569 − 0.569i)13-s + (−0.339 − 0.187i)14-s − 0.588i·15-s + (−0.431 + 0.901i)16-s − 0.0449·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0539i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.998 + 0.0539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.998 + 0.0539i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.998 + 0.0539i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.835249 - 0.0225349i\)
\(L(\frac12)\) \(\approx\) \(0.835249 - 0.0225349i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (28.0 + 15.4i)T \)
5 \( 1 + (-988. - 988. i)T \)
good3 \( 1 + (226. + 226. i)T + 5.90e4iT^{2} \)
7 \( 1 - 6.51e3T + 2.82e8T^{2} \)
11 \( 1 + (-5.06e4 + 5.06e4i)T - 2.59e10iT^{2} \)
13 \( 1 + (-2.11e5 + 2.11e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 6.37e4T + 2.01e12T^{2} \)
19 \( 1 + (-1.28e6 - 1.28e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 3.77e6T + 4.14e13T^{2} \)
29 \( 1 + (1.73e7 - 1.73e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 4.72e7iT - 8.19e14T^{2} \)
37 \( 1 + (2.50e7 + 2.50e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 2.42e7iT - 1.34e16T^{2} \)
43 \( 1 + (-7.75e7 + 7.75e7i)T - 2.16e16iT^{2} \)
47 \( 1 - 8.46e7iT - 5.25e16T^{2} \)
53 \( 1 + (-1.87e8 - 1.87e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (5.48e8 - 5.48e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-6.03e8 + 6.03e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-7.80e8 - 7.80e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 6.21e8T + 3.25e18T^{2} \)
73 \( 1 + 3.36e8iT - 4.29e18T^{2} \)
79 \( 1 - 1.61e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.96e9 - 2.96e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 8.54e9iT - 3.11e19T^{2} \)
97 \( 1 - 2.71e9T + 7.37e19T^{2} \)
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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

−12.10605839306699773841937966154, −11.19391035438017487351596593340, −10.39481540355125392287785460260, −8.946724063897822036007571468063, −7.70496243386656078312957824141, −6.70795019471251775702511781963, −5.62091556827943883959955552154, −3.43064835067315640324822908930, −1.76855170708380718476187165664, −0.909494931424490195611158202307, 0.43701900299118692531188141274, 1.88926355024951910134700178873, 4.30802501195407070356412720028, 5.42808394984687224778655546637, 6.38364341679646283443448029437, 7.88377541877629942745228973218, 9.254529649283719469520726112135, 9.940745374100358424938142004696, 11.12924417247974641024897876121, 11.70981800150618622006493671900

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