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Properties

Label 2-80-5.3-c10-0-16
Degree $2$
Conductor $80$
Sign $0.930 - 0.365i$
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  + (183 + 183i)3-s + (−1.87e3 + 2.50e3i)5-s + (8.40e3 − 8.40e3i)7-s + 7.92e3i·9-s + 1.73e5·11-s + (−2.32e5 − 2.32e5i)13-s + (−8.00e5 + 1.14e5i)15-s + (1.88e6 − 1.88e6i)17-s − 1.10e6i·19-s + 3.07e6·21-s + (5.22e6 + 5.22e6i)23-s + (−2.73e6 − 9.37e6i)25-s + (9.35e6 − 9.35e6i)27-s + 2.47e7i·29-s + 1.00e7·31-s + ⋯
L(s)  = 1  + (0.753 + 0.753i)3-s + (−0.600 + 0.800i)5-s + (0.500 − 0.500i)7-s + 0.134i·9-s + 1.07·11-s + (−0.626 − 0.626i)13-s + (−1.05 + 0.150i)15-s + (1.32 − 1.32i)17-s − 0.444i·19-s + 0.753·21-s + (0.812 + 0.812i)23-s + (−0.280 − 0.960i)25-s + (0.651 − 0.651i)27-s + 1.20i·29-s + 0.351·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.930 - 0.365i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.930 - 0.365i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.81214 + 0.531681i\)
\(L(\frac12)\) \(\approx\) \(2.81214 + 0.531681i\)
\(L(6)\) 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 \( 1 + (1.87e3 - 2.50e3i)T \)
good3 \( 1 + (-183 - 183i)T + 5.90e4iT^{2} \)
7 \( 1 + (-8.40e3 + 8.40e3i)T - 2.82e8iT^{2} \)
11 \( 1 - 1.73e5T + 2.59e10T^{2} \)
13 \( 1 + (2.32e5 + 2.32e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.88e6 + 1.88e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 1.10e6iT - 6.13e12T^{2} \)
23 \( 1 + (-5.22e6 - 5.22e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.47e7iT - 4.20e14T^{2} \)
31 \( 1 - 1.00e7T + 8.19e14T^{2} \)
37 \( 1 + (-5.63e7 + 5.63e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.53e8T + 1.34e16T^{2} \)
43 \( 1 + (5.93e7 + 5.93e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.72e8 - 1.72e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-1.96e8 - 1.96e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 6.94e8iT - 5.11e17T^{2} \)
61 \( 1 - 9.06e8T + 7.13e17T^{2} \)
67 \( 1 + (-9.62e8 + 9.62e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 3.12e9T + 3.25e18T^{2} \)
73 \( 1 + (6.36e8 + 6.36e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 1.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (-5.18e9 - 5.18e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 7.77e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.40e8 - 6.40e8i)T - 7.37e19iT^{2} \)
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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

−12.16902459606303990892942026533, −11.22463385870116491144280032357, −10.06981486691008358423172222584, −9.175832996127858949686686930990, −7.81124047283015470834013005126, −6.91055093572172385871418614281, −4.97775253539517899075786728914, −3.68478938256555617304236131185, −2.91619343759840951912961948598, −0.876589375462448776890139757244, 1.07724503820724280621966867026, 2.02885159368370157468766161578, 3.69656953587398310516044218991, 5.03467745659925098052402753579, 6.66680910871497871461114980247, 8.069005404717339546203960104211, 8.465665380127882376599884672732, 9.790343265996633728105400570922, 11.58853856903694357744293872828, 12.27548829416632039630476343896

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