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Properties

Label 2-80-80.67-c1-0-9
Degree $2$
Conductor $80$
Sign $-0.581 + 0.813i$
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  + (−0.307 − 1.38i)2-s − 2.85i·3-s + (−1.81 + 0.849i)4-s + (1.43 + 1.71i)5-s + (−3.94 + 0.879i)6-s + (−0.458 − 0.458i)7-s + (1.73 + 2.23i)8-s − 5.15·9-s + (1.92 − 2.50i)10-s + (−0.492 − 0.492i)11-s + (2.42 + 5.17i)12-s + 4.52·13-s + (−0.492 + 0.774i)14-s + (4.89 − 4.09i)15-s + (2.55 − 3.07i)16-s + (−3.12 − 3.12i)17-s + ⋯
L(s)  = 1  + (−0.217 − 0.976i)2-s − 1.64i·3-s + (−0.905 + 0.424i)4-s + (0.641 + 0.766i)5-s + (−1.60 + 0.358i)6-s + (−0.173 − 0.173i)7-s + (0.611 + 0.791i)8-s − 1.71·9-s + (0.608 − 0.793i)10-s + (−0.148 − 0.148i)11-s + (0.700 + 1.49i)12-s + 1.25·13-s + (−0.131 + 0.207i)14-s + (1.26 − 1.05i)15-s + (0.638 − 0.769i)16-s + (−0.758 − 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390367 - 0.759155i\)
\(L(\frac12)\) \(\approx\) \(0.390367 - 0.759155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.307 + 1.38i)T \)
5 \( 1 + (-1.43 - 1.71i)T \)
good3 \( 1 + 2.85iT - 3T^{2} \)
7 \( 1 + (0.458 + 0.458i)T + 7iT^{2} \)
11 \( 1 + (0.492 + 0.492i)T + 11iT^{2} \)
13 \( 1 - 4.52T + 13T^{2} \)
17 \( 1 + (3.12 + 3.12i)T + 17iT^{2} \)
19 \( 1 + (-4.04 - 4.04i)T + 19iT^{2} \)
23 \( 1 + (1.80 - 1.80i)T - 23iT^{2} \)
29 \( 1 + (3.83 - 3.83i)T - 29iT^{2} \)
31 \( 1 - 0.139iT - 31T^{2} \)
37 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 + 7.49T + 43T^{2} \)
47 \( 1 + (4.14 - 4.14i)T - 47iT^{2} \)
53 \( 1 + 2.75iT - 53T^{2} \)
59 \( 1 + (3.62 - 3.62i)T - 59iT^{2} \)
61 \( 1 + (-3.72 - 3.72i)T + 61iT^{2} \)
67 \( 1 - 3.32T + 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (2.55 + 2.55i)T + 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (4.95 + 4.95i)T + 97iT^{2} \)
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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

−13.48817234163435190525674148711, −13.17373383567226504267781280765, −11.79204463255198116263366587072, −11.03291486367435442607709788175, −9.662315005026135290119300373323, −8.280073077024364940676002249149, −7.11668157904845414650908814513, −5.83792576967913157050149330710, −3.16493363326819554594006276673, −1.62712533632086963832074203688, 4.03832024927086418200604841773, 5.15757804600645716441508591873, 6.19860244182619703762897077051, 8.367256849495975304396262026000, 9.184930044296836798352740593310, 9.939674245250335959667958867035, 11.08777842613164331845478244036, 13.04569822918566939515944260488, 13.94029945099730860690337448875, 15.21899699083098878131627595399

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