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Properties

Label 2-80-80.27-c1-0-0
Degree $2$
Conductor $80$
Sign $0.669 - 0.743i$
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  + (−1.29 − 0.558i)2-s − 2.55·3-s + (1.37 + 1.45i)4-s + (1.49 + 1.66i)5-s + (3.31 + 1.42i)6-s + (2.40 + 2.40i)7-s + (−0.977 − 2.65i)8-s + 3.51·9-s + (−1.00 − 2.99i)10-s + (−2.67 + 2.67i)11-s + (−3.51 − 3.70i)12-s + 2.40i·13-s + (−1.78 − 4.46i)14-s + (−3.80 − 4.25i)15-s + (−0.212 + 3.99i)16-s + (−0.0750 − 0.0750i)17-s + ⋯
L(s)  = 1  + (−0.918 − 0.394i)2-s − 1.47·3-s + (0.688 + 0.725i)4-s + (0.666 + 0.745i)5-s + (1.35 + 0.581i)6-s + (0.908 + 0.908i)7-s + (−0.345 − 0.938i)8-s + 1.17·9-s + (−0.318 − 0.947i)10-s + (−0.807 + 0.807i)11-s + (−1.01 − 1.06i)12-s + 0.666i·13-s + (−0.475 − 1.19i)14-s + (−0.982 − 1.09i)15-s + (−0.0532 + 0.998i)16-s + (−0.0182 − 0.0182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.669 - 0.743i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431553 + 0.192089i\)
\(L(\frac12)\) \(\approx\) \(0.431553 + 0.192089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.558i)T \)
5 \( 1 + (-1.49 - 1.66i)T \)
good3 \( 1 + 2.55T + 3T^{2} \)
7 \( 1 + (-2.40 - 2.40i)T + 7iT^{2} \)
11 \( 1 + (2.67 - 2.67i)T - 11iT^{2} \)
13 \( 1 - 2.40iT - 13T^{2} \)
17 \( 1 + (0.0750 + 0.0750i)T + 17iT^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 + (3.95 + 3.95i)T + 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 + 2.53iT - 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 + (-2.15 + 2.15i)T - 47iT^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
59 \( 1 + (-5.29 - 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (9.99 + 9.99i)T + 73iT^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 - 5.00i)T + 97iT^{2} \)
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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

−14.82880066365717411730205007013, −13.04076536335767643645371503195, −11.84345523961897809221794657956, −11.25469852264598648329756051107, −10.35009578032724246509833253673, −9.240999362442964191861313462188, −7.54467721628387587324169973087, −6.36608043072745464515277796734, −5.08918913950228620918989300982, −2.19929262790141444212630629229, 1.04722882240148046104196206783, 5.14804413563336529520432990761, 5.75322819075005409568247926644, 7.30880509264319359442398795737, 8.496659254323291188380860269330, 10.10283413170751050193601128324, 10.78179479794496755215010428506, 11.68511204512258079177299261206, 13.09345768293270538062097448962, 14.32850619626756020316718318530

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