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Properties

Label 2-320-20.3-c3-0-2
Degree $2$
Conductor $320$
Sign $-0.405 - 0.913i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.18 − 2.18i)3-s + (−9.23 − 6.30i)5-s + (−6.10 − 6.10i)7-s + 17.4i·9-s − 0.518i·11-s + (−15.4 − 15.4i)13-s + (−33.9 + 6.38i)15-s + (−26.4 + 26.4i)17-s + 28.8·19-s − 26.6·21-s + (−139. + 139. i)23-s + (45.4 + 116. i)25-s + (97.0 + 97.0i)27-s − 160. i·29-s + 279. i·31-s + ⋯
L(s)  = 1  + (0.420 − 0.420i)3-s + (−0.825 − 0.564i)5-s + (−0.329 − 0.329i)7-s + 0.646i·9-s − 0.0142i·11-s + (−0.328 − 0.328i)13-s + (−0.584 + 0.109i)15-s + (−0.377 + 0.377i)17-s + 0.348·19-s − 0.277·21-s + (−1.26 + 1.26i)23-s + (0.363 + 0.931i)25-s + (0.692 + 0.692i)27-s − 1.02i·29-s + 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.405 - 0.913i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ -0.405 - 0.913i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4872026925\)
\(L(\frac12)\) \(\approx\) \(0.4872026925\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.23 + 6.30i)T \)
good3 \( 1 + (-2.18 + 2.18i)T - 27iT^{2} \)
7 \( 1 + (6.10 + 6.10i)T + 343iT^{2} \)
11 \( 1 + 0.518iT - 1.33e3T^{2} \)
13 \( 1 + (15.4 + 15.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (26.4 - 26.4i)T - 4.91e3iT^{2} \)
19 \( 1 - 28.8T + 6.85e3T^{2} \)
23 \( 1 + (139. - 139. i)T - 1.21e4iT^{2} \)
29 \( 1 + 160. iT - 2.43e4T^{2} \)
31 \( 1 - 279. iT - 2.97e4T^{2} \)
37 \( 1 + (63.2 - 63.2i)T - 5.06e4iT^{2} \)
41 \( 1 - 109.T + 6.89e4T^{2} \)
43 \( 1 + (279. - 279. i)T - 7.95e4iT^{2} \)
47 \( 1 + (74.6 + 74.6i)T + 1.03e5iT^{2} \)
53 \( 1 + (-244. - 244. i)T + 1.48e5iT^{2} \)
59 \( 1 + 607.T + 2.05e5T^{2} \)
61 \( 1 + 118.T + 2.26e5T^{2} \)
67 \( 1 + (9.52 + 9.52i)T + 3.00e5iT^{2} \)
71 \( 1 - 860. iT - 3.57e5T^{2} \)
73 \( 1 + (480. + 480. i)T + 3.89e5iT^{2} \)
79 \( 1 - 880.T + 4.93e5T^{2} \)
83 \( 1 + (602. - 602. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 + (1.04e3 - 1.04e3i)T - 9.12e5iT^{2} \)
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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

−11.63854249466240101365629263778, −10.57958007268052080912234639978, −9.580544826279354542133275712217, −8.392820269039741025534675741507, −7.82400429655423945415133755444, −6.93018015235550492189331559063, −5.46761432358791459431359803466, −4.32256637171250957127084868823, −3.13900147828126476632086683026, −1.54441271286045345041384570019, 0.16504988857285309458026749977, 2.50997027765869731723636884903, 3.60048823300896749537347893428, 4.52439774617222629719468338378, 6.13570972602849953537571791668, 7.03777860575557780819810998088, 8.129885709385350121758236450960, 9.072363140466558213130489222928, 9.906930104233925087213617434909, 10.88274733197400202877503341531

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