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Properties

Label 2-320-16.11-c2-0-11
Degree $2$
Conductor $320$
Sign $0.604 + 0.796i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.313 + 0.313i)3-s + (1.58 + 1.58i)5-s − 10.1·7-s − 8.80i·9-s + (6.57 − 6.57i)11-s + (11.4 − 11.4i)13-s + 0.990i·15-s + 24.8·17-s + (−14.7 − 14.7i)19-s + (−3.18 − 3.18i)21-s + 38.0·23-s + 5.00i·25-s + (5.57 − 5.57i)27-s + (−23.6 + 23.6i)29-s + 0.422i·31-s + ⋯
L(s)  = 1  + (0.104 + 0.104i)3-s + (0.316 + 0.316i)5-s − 1.45·7-s − 0.978i·9-s + (0.598 − 0.598i)11-s + (0.880 − 0.880i)13-s + 0.0660i·15-s + 1.46·17-s + (−0.773 − 0.773i)19-s + (−0.151 − 0.151i)21-s + 1.65·23-s + 0.200i·25-s + (0.206 − 0.206i)27-s + (−0.814 + 0.814i)29-s + 0.0136i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37421 - 0.682774i\)
\(L(\frac12)\) \(\approx\) \(1.37421 - 0.682774i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.58 - 1.58i)T \)
good3 \( 1 + (-0.313 - 0.313i)T + 9iT^{2} \)
7 \( 1 + 10.1T + 49T^{2} \)
11 \( 1 + (-6.57 + 6.57i)T - 121iT^{2} \)
13 \( 1 + (-11.4 + 11.4i)T - 169iT^{2} \)
17 \( 1 - 24.8T + 289T^{2} \)
19 \( 1 + (14.7 + 14.7i)T + 361iT^{2} \)
23 \( 1 - 38.0T + 529T^{2} \)
29 \( 1 + (23.6 - 23.6i)T - 841iT^{2} \)
31 \( 1 - 0.422iT - 961T^{2} \)
37 \( 1 + (14.6 + 14.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 44.7iT - 1.68e3T^{2} \)
43 \( 1 + (-31.9 + 31.9i)T - 1.84e3iT^{2} \)
47 \( 1 - 20.4iT - 2.20e3T^{2} \)
53 \( 1 + (38.8 + 38.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-32.2 + 32.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (50.1 - 50.1i)T - 3.72e3iT^{2} \)
67 \( 1 + (42.6 + 42.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 35.4T + 5.04e3T^{2} \)
73 \( 1 + 6.64iT - 5.32e3T^{2} \)
79 \( 1 - 27.2iT - 6.24e3T^{2} \)
83 \( 1 + (-23.2 - 23.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 + 24.5T + 9.40e3T^{2} \)
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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

−11.12198433659084338630569333882, −10.33034957104378156544080754336, −9.299994340876299618682136081971, −8.803477094890157039935565263877, −7.19443654964595688834556111378, −6.36148709558836926037283206109, −5.58363630745595001972674648832, −3.58558556914605946538421807497, −3.13568827576427048375240535864, −0.78799257321868047314777892819, 1.53734579604937433694130973814, 3.13228964145966485550352642110, 4.37408097091459796806122751317, 5.77797451878722103064995634155, 6.61984365635482917624641962328, 7.69550012057341831513922307536, 8.911381373418330926335374539886, 9.644726654555540264465804077003, 10.46673196258065806392144046620, 11.60869966914515763602972822969

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