Properties

Label 2-80-16.13-c1-0-0
Degree $2$
Conductor $80$
Sign $0.929 - 0.367i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.17 − 0.790i)2-s + (1.37 + 1.37i)3-s + (0.750 + 1.85i)4-s + (−0.707 + 0.707i)5-s + (−0.523 − 2.68i)6-s + 2.73i·7-s + (0.584 − 2.76i)8-s + 0.755i·9-s + (1.38 − 0.270i)10-s + (4.12 − 4.12i)11-s + (−1.51 + 3.56i)12-s + (−1.37 − 1.37i)13-s + (2.16 − 3.20i)14-s − 1.93·15-s + (−2.87 + 2.78i)16-s − 4.94·17-s + ⋯
L(s)  = 1  + (−0.829 − 0.558i)2-s + (0.791 + 0.791i)3-s + (0.375 + 0.926i)4-s + (−0.316 + 0.316i)5-s + (−0.213 − 1.09i)6-s + 1.03i·7-s + (0.206 − 0.978i)8-s + 0.251i·9-s + (0.438 − 0.0855i)10-s + (1.24 − 1.24i)11-s + (−0.436 + 1.03i)12-s + (−0.382 − 0.382i)13-s + (0.577 − 0.857i)14-s − 0.500·15-s + (−0.718 + 0.695i)16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.929 - 0.367i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.929 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792314 + 0.151082i\)
\(L(\frac12)\) \(\approx\) \(0.792314 + 0.151082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.790i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-1.37 - 1.37i)T + 3iT^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + (-4.12 + 4.12i)T - 11iT^{2} \)
13 \( 1 + (1.37 + 1.37i)T + 13iT^{2} \)
17 \( 1 + 4.94T + 17T^{2} \)
19 \( 1 + (0.292 + 0.292i)T + 19iT^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 + (5.67 + 5.67i)T + 29iT^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + (-2.48 + 2.48i)T - 37iT^{2} \)
41 \( 1 - 8.40iT - 41T^{2} \)
43 \( 1 + (3.22 - 3.22i)T - 43iT^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 + (-7.20 + 7.20i)T - 53iT^{2} \)
59 \( 1 + (6.41 - 6.41i)T - 59iT^{2} \)
61 \( 1 + (3.82 + 3.82i)T + 61iT^{2} \)
67 \( 1 + (-5.76 - 5.76i)T + 67iT^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + (0.516 + 0.516i)T + 83iT^{2} \)
89 \( 1 - 6.42iT - 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−14.76986539425492041855302247666, −13.34772620623547195091417190730, −11.86173398960946792684976253058, −11.19000455020586103077216171552, −9.732519834426879671420778363612, −8.977760513209140237208351944162, −8.190435306715580283801380366522, −6.39835206876916354259853669460, −3.96753835173743538524118515468, −2.75379488932940346509866388665, 1.76492294093804938746179044531, 4.47585443271381284196301224819, 6.84426869543393618934015711383, 7.29570378865105828726238381266, 8.566540071455203923156655452999, 9.509374994916072572143186310945, 10.84993919754072972700555144728, 12.23253159214325167159078213146, 13.54355396621173847692855270362, 14.39991178755739787337809902138

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