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Properties

Label 2-320-320.3-c1-0-13
Degree $2$
Conductor $320$
Sign $0.805 + 0.592i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.171 − 1.40i)2-s + (0.198 + 0.296i)3-s + (−1.94 − 0.481i)4-s + (−0.936 + 2.03i)5-s + (0.450 − 0.227i)6-s + (3.61 + 1.49i)7-s + (−1.00 + 2.64i)8-s + (1.09 − 2.65i)9-s + (2.69 + 1.66i)10-s + (−0.389 − 1.95i)11-s + (−0.241 − 0.670i)12-s + (4.89 + 0.973i)13-s + (2.72 − 4.82i)14-s + (−0.787 + 0.124i)15-s + (3.53 + 1.86i)16-s + 0.690·17-s + ⋯
L(s)  = 1  + (0.121 − 0.992i)2-s + (0.114 + 0.171i)3-s + (−0.970 − 0.240i)4-s + (−0.418 + 0.908i)5-s + (0.183 − 0.0927i)6-s + (1.36 + 0.566i)7-s + (−0.356 + 0.934i)8-s + (0.366 − 0.884i)9-s + (0.850 + 0.525i)10-s + (−0.117 − 0.590i)11-s + (−0.0698 − 0.193i)12-s + (1.35 + 0.269i)13-s + (0.727 − 1.28i)14-s + (−0.203 + 0.0321i)15-s + (0.884 + 0.466i)16-s + 0.167·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38019 - 0.453135i\)
\(L(\frac12)\) \(\approx\) \(1.38019 - 0.453135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.171 + 1.40i)T \)
5 \( 1 + (0.936 - 2.03i)T \)
good3 \( 1 + (-0.198 - 0.296i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (-3.61 - 1.49i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.389 + 1.95i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-4.89 - 0.973i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 - 0.690T + 17T^{2} \)
19 \( 1 + (-2.43 - 3.63i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (1.84 + 4.46i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.05 - 5.31i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 0.845T + 31T^{2} \)
37 \( 1 + (-1.38 - 6.95i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.34 - 8.07i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (10.1 + 6.76i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 + (-1.81 - 1.21i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (-3.38 - 2.25i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-1.60 + 8.06i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (2.28 - 1.52i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (0.739 + 1.78i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.25 - 5.45i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.0950 - 0.0950i)T - 79iT^{2} \)
83 \( 1 + (12.9 + 2.58i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (1.92 - 0.798i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (2.73 + 2.73i)T + 97iT^{2} \)
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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

−11.64362249707257987647820816626, −10.75743730066434701783197912632, −9.984366755906405468807890323522, −8.618305029660438716440496928072, −8.220472218345082237779781339569, −6.55525731123624619307978357834, −5.36237152511592596658030565155, −4.01635933455748604580503293056, −3.18625507897787604926013241346, −1.55338400676644497246942192871, 1.36391445961442311636308310399, 3.97534580084546500184099302131, 4.78659197312355949219405748994, 5.61520207614903578308819200981, 7.26972659156458577367700287373, 7.86278689190472657823460760462, 8.477729343700074954752874936412, 9.590670811466566782924372953926, 10.87372069787895712096274540192, 11.77711935878910641132957453661

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