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Properties

Label 2-80-80.3-c1-0-4
Degree $2$
Conductor $80$
Sign $0.999 - 0.0200i$
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  + (−1.38 + 0.307i)2-s + 2.85·3-s + (1.81 − 0.849i)4-s + (−1.71 − 1.43i)5-s + (−3.94 + 0.879i)6-s + (−0.458 + 0.458i)7-s + (−2.23 + 1.73i)8-s + 5.15·9-s + (2.80 + 1.45i)10-s + (−0.492 − 0.492i)11-s + (5.17 − 2.42i)12-s + 4.52i·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  + (−0.976 + 0.217i)2-s + 1.64·3-s + (0.905 − 0.424i)4-s + (−0.766 − 0.641i)5-s + (−1.60 + 0.358i)6-s + (−0.173 + 0.173i)7-s + (−0.791 + 0.611i)8-s + 1.71·9-s + (0.888 + 0.459i)10-s + (−0.148 − 0.148i)11-s + (1.49 − 0.700i)12-s + 1.25i·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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.999 - 0.0200i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.999 - 0.0200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898087 + 0.00899030i\)
\(L(\frac12)\) \(\approx\) \(0.898087 + 0.00899030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−14.70163809646811249694947051059, −13.49726666599547191693869829338, −12.24657568654827227753437136085, −10.87355768066164716680751495856, −9.342009555296052050996887091755, −8.753160246570514032170362482883, −7.969638826176142879567816883993, −6.71516230036312236492763649200, −4.16619993438205904243084955684, −2.28443127955614854008954789119, 2.58202536610785759150904979431, 3.68481460562402221746409382995, 6.86453037199916957470210154737, 7.901649370793517205816128890697, 8.534002622240983379271008313472, 9.864848918398146468630431555918, 10.70895348002835505541852470220, 12.19500905750124686572094158962, 13.36529029365862882527797706377, 14.70995917268749808985698972704

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