Properties

Label 2-80-5.3-c4-0-4
Degree $2$
Conductor $80$
Sign $0.170 - 0.985i$
Analytic cond. $8.26959$
Root an. cond. $2.87569$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.3 + 11.3i)3-s + (9.33 + 23.1i)5-s + (42.8 − 42.8i)7-s + 175. i·9-s + 20.3·11-s + (−151. − 151. i)13-s + (−157. + 368. i)15-s + (−150. + 150. i)17-s − 277. i·19-s + 971.·21-s + (363. + 363. i)23-s + (−450. + 432. i)25-s + (−1.07e3 + 1.07e3i)27-s − 413. i·29-s + 659.·31-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)3-s + (0.373 + 0.927i)5-s + (0.874 − 0.874i)7-s + 2.16i·9-s + 0.168·11-s + (−0.897 − 0.897i)13-s + (−0.698 + 1.63i)15-s + (−0.519 + 0.519i)17-s − 0.767i·19-s + 2.20·21-s + (0.687 + 0.687i)23-s + (−0.721 + 0.692i)25-s + (−1.47 + 1.47i)27-s − 0.491i·29-s + 0.686·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(8.26959\)
Root analytic conductor: \(2.87569\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :2),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.04964 + 1.72586i\)
\(L(\frac12)\) \(\approx\) \(2.04964 + 1.72586i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-9.33 - 23.1i)T \)
good3 \( 1 + (-11.3 - 11.3i)T + 81iT^{2} \)
7 \( 1 + (-42.8 + 42.8i)T - 2.40e3iT^{2} \)
11 \( 1 - 20.3T + 1.46e4T^{2} \)
13 \( 1 + (151. + 151. i)T + 2.85e4iT^{2} \)
17 \( 1 + (150. - 150. i)T - 8.35e4iT^{2} \)
19 \( 1 + 277. iT - 1.30e5T^{2} \)
23 \( 1 + (-363. - 363. i)T + 2.79e5iT^{2} \)
29 \( 1 + 413. iT - 7.07e5T^{2} \)
31 \( 1 - 659.T + 9.23e5T^{2} \)
37 \( 1 + (-1.05e3 + 1.05e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 801.T + 2.82e6T^{2} \)
43 \( 1 + (2.00e3 + 2.00e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.47e3 + 1.47e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (1.32e3 + 1.32e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 937. iT - 1.21e7T^{2} \)
61 \( 1 - 2.54e3T + 1.38e7T^{2} \)
67 \( 1 + (140. - 140. i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.53e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.90e3 - 5.90e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 196. iT - 3.89e7T^{2} \)
83 \( 1 + (-9.13e3 - 9.13e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.06e4iT - 6.27e7T^{2} \)
97 \( 1 + (1.10e4 - 1.10e4i)T - 8.85e7iT^{2} \)
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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

−14.10357208241997986810900448521, −13.36289128978468455606901120371, −11.18980611790922831857228056774, −10.40735629086616817327841613112, −9.598885608049381037745495210713, −8.271534329619365525705340067880, −7.21995871503328386340181438344, −5.02807165300085912270094546932, −3.75074190092590491427965348022, −2.43426118574892713174599489478, 1.44331795596448514270834584199, 2.50344739791039117738062131833, 4.77605734363693899400104911506, 6.51696271820082358888932566588, 7.888671209380698122239687971767, 8.710535260121886977930962296467, 9.481572850873519034316570820954, 11.77293929024967944845200071432, 12.42964590945176633879996899797, 13.45196005461261320077720795765

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