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Properties

Label 2-320-320.27-c1-0-11
Degree $2$
Conductor $320$
Sign $-0.617 - 0.786i$
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  + (0.951 + 1.04i)2-s + (0.397 − 0.0790i)3-s + (−0.190 + 1.99i)4-s + (−1.72 + 1.42i)5-s + (0.460 + 0.340i)6-s + (0.0192 + 0.0463i)7-s + (−2.26 + 1.69i)8-s + (−2.61 + 1.08i)9-s + (−3.13 − 0.442i)10-s + (1.80 + 2.69i)11-s + (0.0816 + 0.806i)12-s + (−0.557 − 0.372i)13-s + (−0.0302 + 0.0642i)14-s + (−0.571 + 0.703i)15-s + (−3.92 − 0.758i)16-s + 6.76·17-s + ⋯
L(s)  = 1  + (0.672 + 0.740i)2-s + (0.229 − 0.0456i)3-s + (−0.0952 + 0.995i)4-s + (−0.769 + 0.638i)5-s + (0.188 + 0.139i)6-s + (0.00726 + 0.0175i)7-s + (−0.800 + 0.599i)8-s + (−0.873 + 0.361i)9-s + (−0.990 − 0.140i)10-s + (0.543 + 0.812i)11-s + (0.0235 + 0.232i)12-s + (−0.154 − 0.103i)13-s + (−0.00809 + 0.0171i)14-s + (−0.147 + 0.181i)15-s + (−0.981 − 0.189i)16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694652 + 1.42820i\)
\(L(\frac12)\) \(\approx\) \(0.694652 + 1.42820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 1.04i)T \)
5 \( 1 + (1.72 - 1.42i)T \)
good3 \( 1 + (-0.397 + 0.0790i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.0192 - 0.0463i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.80 - 2.69i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (0.557 + 0.372i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 - 6.76T + 17T^{2} \)
19 \( 1 + (-1.93 + 0.383i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (-2.67 - 1.10i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.31 + 3.46i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (-1.91 - 2.86i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-5.65 + 2.34i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.88 - 9.45i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + 6.03iT - 47T^{2} \)
53 \( 1 + (-0.281 + 1.41i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (-0.372 + 1.87i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (2.68 - 4.02i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (-1.61 - 8.11i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (6.04 + 2.50i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.57 + 0.651i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.39 + 6.39i)T + 79iT^{2} \)
83 \( 1 + (-4.60 - 3.07i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (-0.439 + 1.06i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (-10.0 + 10.0i)T - 97iT^{2} \)
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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

−11.91684234675876207024448397714, −11.46916896973815531726454420800, −10.08725960431015927734488094498, −8.861254163289145769702578406934, −7.79181646553815604986716094633, −7.31744404331136041347046663650, −6.10803535636346560778850793926, −5.01282191887267340776912319344, −3.74582814403744040045821817243, −2.78375197714732389772270344332, 0.971160547826260075135854438500, 3.06313103052642422382666628479, 3.82547520781622127574541939871, 5.16355086618873358009983340301, 6.02051280991280925737365086176, 7.53812059514431938790131324037, 8.753116248666712929246512708637, 9.361282673103409436171410594447, 10.66338145935265810029477175782, 11.54658059669251766535097654749

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