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Properties

Label 2-320-16.3-c2-0-15
Degree $2$
Conductor $320$
Sign $-0.829 + 0.557i$
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  + (2.05 − 2.05i)3-s + (1.58 − 1.58i)5-s − 10.3·7-s + 0.519i·9-s + (−14.0 − 14.0i)11-s + (−9.77 − 9.77i)13-s − 6.51i·15-s + 7.10·17-s + (16.6 − 16.6i)19-s + (−21.4 + 21.4i)21-s − 24.8·23-s − 5.00i·25-s + (19.6 + 19.6i)27-s + (11.2 + 11.2i)29-s − 7.04i·31-s + ⋯
L(s)  = 1  + (0.686 − 0.686i)3-s + (0.316 − 0.316i)5-s − 1.48·7-s + 0.0576i·9-s + (−1.27 − 1.27i)11-s + (−0.751 − 0.751i)13-s − 0.434i·15-s + 0.417·17-s + (0.874 − 0.874i)19-s + (−1.01 + 1.01i)21-s − 1.07·23-s − 0.200i·25-s + (0.726 + 0.726i)27-s + (0.389 + 0.389i)29-s − 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.829 + 0.557i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.829 + 0.557i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.333263 - 1.09315i\)
\(L(\frac12)\) \(\approx\) \(0.333263 - 1.09315i\)
\(L(2)\) 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 \( 1 + (-2.05 + 2.05i)T - 9iT^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + (14.0 + 14.0i)T + 121iT^{2} \)
13 \( 1 + (9.77 + 9.77i)T + 169iT^{2} \)
17 \( 1 - 7.10T + 289T^{2} \)
19 \( 1 + (-16.6 + 16.6i)T - 361iT^{2} \)
23 \( 1 + 24.8T + 529T^{2} \)
29 \( 1 + (-11.2 - 11.2i)T + 841iT^{2} \)
31 \( 1 + 7.04iT - 961T^{2} \)
37 \( 1 + (-11.2 + 11.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 62.6iT - 1.68e3T^{2} \)
43 \( 1 + (-37.0 - 37.0i)T + 1.84e3iT^{2} \)
47 \( 1 + 0.176iT - 2.20e3T^{2} \)
53 \( 1 + (32.3 - 32.3i)T - 2.80e3iT^{2} \)
59 \( 1 + (25.8 + 25.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (36.8 + 36.8i)T + 3.72e3iT^{2} \)
67 \( 1 + (-12.8 + 12.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 64.6T + 5.04e3T^{2} \)
73 \( 1 + 19.2iT - 5.32e3T^{2} \)
79 \( 1 - 48.9iT - 6.24e3T^{2} \)
83 \( 1 + (-18.5 + 18.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 43.4iT - 7.92e3T^{2} \)
97 \( 1 + 115.T + 9.40e3T^{2} \)
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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

−10.88460572419179190162204560888, −10.00973964275247819921783406647, −9.131470045751148029326261546587, −8.065713698206038211786760072955, −7.39002817847374732127208265908, −6.10444458571768604512947718107, −5.21401296904855224591679670669, −3.23021190602546380476004571852, −2.56518132841063055841028981692, −0.46240033261669421673792382693, 2.41175524211877586835213381806, 3.36695125951056116680495115687, 4.56061134728910341705988977536, 5.94130713562563246247286724916, 7.01756985114569739179042864464, 7.992968557775835323856742262701, 9.489176570920332078677232484923, 9.795101381769692554836701494405, 10.31638036631070956005025665997, 12.02811638428556674213286280831

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