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Properties

Label 2-320-320.307-c1-0-45
Degree $2$
Conductor $320$
Sign $-0.661 - 0.749i$
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  + (0.529 − 1.31i)2-s + (−0.658 − 0.131i)3-s + (−1.43 − 1.38i)4-s + (−1.83 + 1.27i)5-s + (−0.520 + 0.794i)6-s + (0.475 + 0.196i)7-s + (−2.58 + 1.15i)8-s + (−2.35 − 0.975i)9-s + (0.699 + 3.08i)10-s + (−2.23 − 1.49i)11-s + (0.766 + 1.10i)12-s + (−3.63 + 2.42i)13-s + (0.510 − 0.519i)14-s + (1.37 − 0.599i)15-s + (0.143 + 3.99i)16-s − 5.64i·17-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.380 − 0.0756i)3-s + (−0.719 − 0.694i)4-s + (−0.821 + 0.570i)5-s + (−0.212 + 0.324i)6-s + (0.179 + 0.0744i)7-s + (−0.913 + 0.407i)8-s + (−0.784 − 0.325i)9-s + (0.221 + 0.975i)10-s + (−0.672 − 0.449i)11-s + (0.221 + 0.318i)12-s + (−1.00 + 0.672i)13-s + (0.136 − 0.138i)14-s + (0.355 − 0.154i)15-s + (0.0359 + 0.999i)16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0693633 + 0.153800i\)
\(L(\frac12)\) \(\approx\) \(0.0693633 + 0.153800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.529 + 1.31i)T \)
5 \( 1 + (1.83 - 1.27i)T \)
good3 \( 1 + (0.658 + 0.131i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (-0.475 - 0.196i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.23 + 1.49i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (3.63 - 2.42i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + 5.64iT - 17T^{2} \)
19 \( 1 + (1.06 - 5.37i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.502 - 1.21i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.77 + 2.51i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + 6.11T + 31T^{2} \)
37 \( 1 + (-3.73 + 5.58i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (7.32 + 3.03i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.834 + 4.19i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 - 5.15T + 47T^{2} \)
53 \( 1 + (0.827 + 4.16i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (10.3 - 2.06i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (9.04 - 6.04i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.39 - 7.00i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (-14.5 + 6.02i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.132 - 0.320i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (11.1 - 11.1i)T - 79iT^{2} \)
83 \( 1 + (-3.27 + 2.18i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-0.852 - 2.05i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (0.330 - 0.330i)T - 97iT^{2} \)
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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

−11.25056819231550758002773830949, −10.44361211834109005130370097447, −9.386744254813442208641864516437, −8.303778980148078008751187858721, −7.13618768893874875856589617029, −5.82843678121615240126395782289, −4.84747123117292483066754851195, −3.55685659696342331529851022641, −2.48102368023939653652303215291, −0.10634188423767094689403554227, 3.04513177877502555110002499166, 4.64259591111139434903557048358, 5.08185556149659965899097838269, 6.33531784416872518033344877951, 7.58812406830846684682188092179, 8.157540256293373642031216207791, 9.082977133392622111847284515485, 10.45343594153219646239771949374, 11.44191835737699919973808560619, 12.54560765106448944401566597502

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