Properties

Label 2-320-5.2-c4-0-40
Degree $2$
Conductor $320$
Sign $-0.871 + 0.489i$
Analytic cond. $33.0783$
Root an. cond. $5.75138$
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  + (−0.958 + 0.958i)3-s + (1.04 − 24.9i)5-s + (−26.0 − 26.0i)7-s + 79.1i·9-s + 75.9·11-s + (99.1 − 99.1i)13-s + (22.9 + 24.9i)15-s + (77.9 + 77.9i)17-s − 640. i·19-s + 49.9·21-s + (−117. + 117. i)23-s + (−622. − 52.0i)25-s + (−153. − 153. i)27-s + 752. i·29-s − 708.·31-s + ⋯
L(s)  = 1  + (−0.106 + 0.106i)3-s + (0.0416 − 0.999i)5-s + (−0.531 − 0.531i)7-s + 0.977i·9-s + 0.627·11-s + (0.586 − 0.586i)13-s + (0.101 + 0.110i)15-s + (0.269 + 0.269i)17-s − 1.77i·19-s + 0.113·21-s + (−0.221 + 0.221i)23-s + (−0.996 − 0.0832i)25-s + (−0.210 − 0.210i)27-s + 0.894i·29-s − 0.736·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.871 + 0.489i$
Analytic conductor: \(33.0783\)
Root analytic conductor: \(5.75138\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :2),\ -0.871 + 0.489i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9306874056\)
\(L(\frac12)\) \(\approx\) \(0.9306874056\)
\(L(3)\) 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 \( 1 + (-1.04 + 24.9i)T \)
good3 \( 1 + (0.958 - 0.958i)T - 81iT^{2} \)
7 \( 1 + (26.0 + 26.0i)T + 2.40e3iT^{2} \)
11 \( 1 - 75.9T + 1.46e4T^{2} \)
13 \( 1 + (-99.1 + 99.1i)T - 2.85e4iT^{2} \)
17 \( 1 + (-77.9 - 77.9i)T + 8.35e4iT^{2} \)
19 \( 1 + 640. iT - 1.30e5T^{2} \)
23 \( 1 + (117. - 117. i)T - 2.79e5iT^{2} \)
29 \( 1 - 752. iT - 7.07e5T^{2} \)
31 \( 1 + 708.T + 9.23e5T^{2} \)
37 \( 1 + (1.17e3 + 1.17e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.82e3T + 2.82e6T^{2} \)
43 \( 1 + (2.37e3 - 2.37e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (2.45e3 + 2.45e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (2.11e3 - 2.11e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 5.25e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.31e3T + 1.38e7T^{2} \)
67 \( 1 + (5.56e3 + 5.56e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 7.55e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.20e3 + 1.20e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 3.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (-928. + 928. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.19e4iT - 6.27e7T^{2} \)
97 \( 1 + (-4.68e3 - 4.68e3i)T + 8.85e7iT^{2} \)
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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

−10.66762433428455056484733333643, −9.603647057367717410257283172892, −8.780879534624701127697857027081, −7.81502724028188313839693521963, −6.71231850684872607584859558296, −5.46670678495784481598680369399, −4.59520974999496670089042354274, −3.37330577818421323619210386328, −1.62636229568627235937896148336, −0.28282746720188618620603907111, 1.61468353230523660160134059211, 3.16222809715400969671459912956, 3.97092640363005096581629637405, 5.93901015837692072669074771308, 6.33526883603950045669878128767, 7.39081377561240352592341979643, 8.673402361054075475966972136453, 9.616584702183674014661860752117, 10.34161228674725491352567195910, 11.65360779829948486211058118280

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