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Properties

Label 2-80-5.2-c10-0-0
Degree $2$
Conductor $80$
Sign $-0.981 + 0.192i$
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  + (−57 + 57i)3-s + (2.92e3 + 1.10e3i)5-s + (−6.95e3 − 6.95e3i)7-s + 5.25e4i·9-s − 7.52e4·11-s + (1.09e5 − 1.09e5i)13-s + (−2.29e5 + 1.04e5i)15-s + (−1.52e6 − 1.52e6i)17-s + 4.03e6i·19-s + 7.92e5·21-s + (7.12e5 − 7.12e5i)23-s + (7.34e6 + 6.43e6i)25-s + (−6.36e6 − 6.36e6i)27-s + 4.46e5i·29-s + 2.90e7·31-s + ⋯
L(s)  = 1  + (−0.234 + 0.234i)3-s + (0.936 + 0.352i)5-s + (−0.413 − 0.413i)7-s + 0.889i·9-s − 0.467·11-s + (0.295 − 0.295i)13-s + (−0.302 + 0.136i)15-s + (−1.07 − 1.07i)17-s + 1.63i·19-s + 0.194·21-s + (0.110 − 0.110i)23-s + (0.752 + 0.658i)25-s + (−0.443 − 0.443i)27-s + 0.0217i·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.981 + 0.192i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.981 + 0.192i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0292204 - 0.300503i\)
\(L(\frac12)\) \(\approx\) \(0.0292204 - 0.300503i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.92e3 - 1.10e3i)T \)
good3 \( 1 + (57 - 57i)T - 5.90e4iT^{2} \)
7 \( 1 + (6.95e3 + 6.95e3i)T + 2.82e8iT^{2} \)
11 \( 1 + 7.52e4T + 2.59e10T^{2} \)
13 \( 1 + (-1.09e5 + 1.09e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.52e6 + 1.52e6i)T + 2.01e12iT^{2} \)
19 \( 1 - 4.03e6iT - 6.13e12T^{2} \)
23 \( 1 + (-7.12e5 + 7.12e5i)T - 4.14e13iT^{2} \)
29 \( 1 - 4.46e5iT - 4.20e14T^{2} \)
31 \( 1 - 2.90e7T + 8.19e14T^{2} \)
37 \( 1 + (9.11e5 + 9.11e5i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.63e8T + 1.34e16T^{2} \)
43 \( 1 + (1.18e8 - 1.18e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (2.76e8 + 2.76e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (-3.08e8 + 3.08e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 9.40e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.35e9T + 7.13e17T^{2} \)
67 \( 1 + (8.53e8 + 8.53e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 2.82e9T + 3.25e18T^{2} \)
73 \( 1 + (2.75e9 - 2.75e9i)T - 4.29e18iT^{2} \)
79 \( 1 + 3.32e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.34e9 - 1.34e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 2.66e9iT - 3.11e19T^{2} \)
97 \( 1 + (5.26e8 + 5.26e8i)T + 7.37e19iT^{2} \)
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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

−13.29598719205831613579587599457, −11.66297651845619897387165355647, −10.41762550556748450511996328853, −9.979963430084759142546454458985, −8.438942074128752767890051530048, −7.05822344503519082533989008102, −5.87769914420638975320917252980, −4.74740894537349109309589920108, −3.04094953893898049213215259336, −1.73973951116215075848460140511, 0.07634144617562274761708078176, 1.52514604638724406305652323284, 2.88532084213068732188294604006, 4.65683492308192231838892134885, 6.03853689554902176957595624925, 6.73863381912348125453826362737, 8.637018490890553569405016508802, 9.374393319673714589192203984971, 10.60772839179487624517335820691, 11.86418977134853957236612255799

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