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Properties

Label 2-80-16.5-c1-0-7
Degree $2$
Conductor $80$
Sign $-0.0988 + 0.995i$
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  + (−0.376 − 1.36i)2-s + (1.82 − 1.82i)3-s + (−1.71 + 1.02i)4-s + (−0.707 − 0.707i)5-s + (−3.18 − 1.80i)6-s + 4.50i·7-s + (2.04 + 1.95i)8-s − 3.68i·9-s + (−0.697 + 1.23i)10-s + (−1.64 − 1.64i)11-s + (−1.25 + 5.01i)12-s + (1.51 − 1.51i)13-s + (6.14 − 1.69i)14-s − 2.58·15-s + (1.88 − 3.52i)16-s + 1.45·17-s + ⋯
L(s)  = 1  + (−0.266 − 0.963i)2-s + (1.05 − 1.05i)3-s + (−0.857 + 0.513i)4-s + (−0.316 − 0.316i)5-s + (−1.29 − 0.735i)6-s + 1.70i·7-s + (0.723 + 0.689i)8-s − 1.22i·9-s + (−0.220 + 0.389i)10-s + (−0.494 − 0.494i)11-s + (−0.363 + 1.44i)12-s + (0.421 − 0.421i)13-s + (1.64 − 0.454i)14-s − 0.667·15-s + (0.472 − 0.881i)16-s + 0.353·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0988 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0988 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.0988 + 0.995i$
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: \((2,\ 80,\ (\ :1/2),\ -0.0988 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666421 - 0.735884i\)
\(L(\frac12)\) \(\approx\) \(0.666421 - 0.735884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.376 + 1.36i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.82 + 1.82i)T - 3iT^{2} \)
7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 + (1.64 + 1.64i)T + 11iT^{2} \)
13 \( 1 + (-1.51 + 1.51i)T - 13iT^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 + (2.67 - 2.67i)T - 19iT^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
29 \( 1 + (-0.924 + 0.924i)T - 29iT^{2} \)
31 \( 1 + 7.20T + 31T^{2} \)
37 \( 1 + (5.21 + 5.21i)T + 37iT^{2} \)
41 \( 1 + 6.41iT - 41T^{2} \)
43 \( 1 + (-7.65 - 7.65i)T + 43iT^{2} \)
47 \( 1 + 2.51T + 47T^{2} \)
53 \( 1 + (-1.50 - 1.50i)T + 53iT^{2} \)
59 \( 1 + (5.31 + 5.31i)T + 59iT^{2} \)
61 \( 1 + (1.02 - 1.02i)T - 61iT^{2} \)
67 \( 1 + (-5.22 + 5.22i)T - 67iT^{2} \)
71 \( 1 + 1.92iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 - 9.36iT - 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−13.81233730874694649503616323634, −12.66316755462734607074089909619, −12.37571028843833352734430446061, −11.01396473911854365108775664753, −9.269412083593274534852182304164, −8.527250540411935600621316309651, −7.78351584323572552141405667020, −5.60535764418082492606475597258, −3.29575440362236288092380560256, −2.03022076746033015488101457574, 3.73755730382643648849698937949, 4.65335156379218792221861664415, 6.86794620573944544649623891067, 7.84126496234031060278032406163, 8.970342272924689566178977452773, 10.13652678341726017087153635563, 10.72763045386641419368718933036, 13.14877535830764258062259869839, 14.06951415423938629829684853257, 14.71796274625662716417148016335

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