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Properties

Label 2-80-16.5-c1-0-2
Degree $2$
Conductor $80$
Sign $0.704 - 0.709i$
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  + (1.40 − 0.139i)2-s + (−2.32 + 2.32i)3-s + (1.96 − 0.393i)4-s + (0.707 + 0.707i)5-s + (−2.94 + 3.59i)6-s − 0.982i·7-s + (2.70 − 0.828i)8-s − 7.82i·9-s + (1.09 + 0.896i)10-s + (−1.62 − 1.62i)11-s + (−3.64 + 5.47i)12-s + (−0.690 + 0.690i)13-s + (−0.137 − 1.38i)14-s − 3.28·15-s + (3.68 − 1.54i)16-s − 2.19·17-s + ⋯
L(s)  = 1  + (0.995 − 0.0989i)2-s + (−1.34 + 1.34i)3-s + (0.980 − 0.196i)4-s + (0.316 + 0.316i)5-s + (−1.20 + 1.46i)6-s − 0.371i·7-s + (0.956 − 0.292i)8-s − 2.60i·9-s + (0.345 + 0.283i)10-s + (−0.490 − 0.490i)11-s + (−1.05 + 1.58i)12-s + (−0.191 + 0.191i)13-s + (−0.0367 − 0.369i)14-s − 0.849·15-s + (0.922 − 0.386i)16-s − 0.532·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08283 + 0.450920i\)
\(L(\frac12)\) \(\approx\) \(1.08283 + 0.450920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.139i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (2.32 - 2.32i)T - 3iT^{2} \)
7 \( 1 + 0.982iT - 7T^{2} \)
11 \( 1 + (1.62 + 1.62i)T + 11iT^{2} \)
13 \( 1 + (0.690 - 0.690i)T - 13iT^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 + (-1.92 + 1.92i)T - 19iT^{2} \)
23 \( 1 - 2.01iT - 23T^{2} \)
29 \( 1 + (5.27 - 5.27i)T - 29iT^{2} \)
31 \( 1 - 0.435T + 31T^{2} \)
37 \( 1 + (5.79 + 5.79i)T + 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (0.507 + 0.507i)T + 43iT^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 + (-6.29 - 6.29i)T + 53iT^{2} \)
59 \( 1 + (5.67 + 5.67i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (-4.53 + 4.53i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 9.24iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (0.683 - 0.683i)T - 83iT^{2} \)
89 \( 1 - 5.44iT - 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−14.69956755543307108625693095900, −13.47087845790127091292922242231, −12.17533942547689782465379249828, −11.09502417809936256461583186745, −10.64853230540270707772625176351, −9.469543948878706704398319418765, −6.96658056239190043212446974780, −5.77409043113278268502091282870, −4.86884280850687598442586466180, −3.53145371421731720630799622830, 2.05849018613785828681745526692, 4.94254582513974706666841119238, 5.82776280358360560059081917157, 6.87963040737156692782462460964, 7.962036844478056711588190138421, 10.36058473490034610161954081099, 11.52577395144196121450121546647, 12.27502214178384176036045193140, 13.03298173956136939111099711811, 13.77716184960941211050112420459

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