Properties

Label 2-80-80.29-c5-0-19
Degree $2$
Conductor $80$
Sign $0.137 - 0.990i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.03 + 3.96i)2-s + (−19.4 + 19.4i)3-s + (0.557 − 31.9i)4-s + (−49.1 − 26.6i)5-s + (1.35 − 155. i)6-s + 138.·7-s + (124. + 131. i)8-s − 516. i·9-s + (303. − 87.5i)10-s + (148. − 148. i)11-s + (612. + 634. i)12-s + (−395. + 395. i)13-s + (−556. + 547. i)14-s + (1.47e3 − 439. i)15-s + (−1.02e3 − 35.6i)16-s − 1.34e3i·17-s + ⋯
L(s)  = 1  + (−0.713 + 0.700i)2-s + (−1.25 + 1.25i)3-s + (0.0174 − 0.999i)4-s + (−0.879 − 0.476i)5-s + (0.0154 − 1.76i)6-s + 1.06·7-s + (0.688 + 0.725i)8-s − 2.12i·9-s + (0.960 − 0.276i)10-s + (0.370 − 0.370i)11-s + (1.22 + 1.27i)12-s + (−0.649 + 0.649i)13-s + (−0.759 + 0.746i)14-s + (1.69 − 0.504i)15-s + (−0.999 − 0.0348i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.137 - 0.990i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.461460 + 0.401672i\)
\(L(\frac12)\) \(\approx\) \(0.461460 + 0.401672i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.03 - 3.96i)T \)
5 \( 1 + (49.1 + 26.6i)T \)
good3 \( 1 + (19.4 - 19.4i)T - 243iT^{2} \)
7 \( 1 - 138.T + 1.68e4T^{2} \)
11 \( 1 + (-148. + 148. i)T - 1.61e5iT^{2} \)
13 \( 1 + (395. - 395. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.34e3iT - 1.41e6T^{2} \)
19 \( 1 + (338. + 338. i)T + 2.47e6iT^{2} \)
23 \( 1 + 3.13e3T + 6.43e6T^{2} \)
29 \( 1 + (-3.34e3 - 3.34e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 8.97e3T + 2.86e7T^{2} \)
37 \( 1 + (-3.91e3 - 3.91e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.15e4iT - 1.15e8T^{2} \)
43 \( 1 + (-9.09e3 - 9.09e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 2.31e4iT - 2.29e8T^{2} \)
53 \( 1 + (4.38e3 + 4.38e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.01e4 + 1.01e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (7.48e3 + 7.48e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (4.13e4 - 4.13e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.53e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.19e4T + 2.07e9T^{2} \)
79 \( 1 - 8.35e4T + 3.07e9T^{2} \)
83 \( 1 + (-5.71e4 + 5.71e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 4.85e4iT - 5.58e9T^{2} \)
97 \( 1 - 8.51e4iT - 8.58e9T^{2} \)
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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

−14.19521822983216727196256785895, −11.74560387160126348997240538317, −11.53214121962505152652881818239, −10.27972462894843715235951550658, −9.220831488649752626779190923701, −8.040119802654521871415713067582, −6.52918117294229411367128251757, −5.02397984859385782297109473021, −4.47801856252244226195044363243, −0.72671963479849813690821488653, 0.68426844635927948292813947506, 2.10733875084569150819197487859, 4.42397184750505561582564192882, 6.33109794108098265688086187122, 7.66094793019462836827801808275, 8.091092677169149406161910344737, 10.36678842009343820733498809470, 11.10523721426566059381918446920, 12.18274683865089466954363762100, 12.30969179398480164624898931222

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