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Properties

Label 2-320-5.4-c5-0-40
Degree $2$
Conductor $320$
Sign $0.825 + 0.564i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.01i·3-s + (46.1 + 31.5i)5-s − 15.3i·7-s + 217.·9-s + 576.·11-s − 607. i·13-s + (158. − 231. i)15-s − 2.01e3i·17-s − 2.42e3·19-s − 77.2·21-s + 4.35e3i·23-s + (1.13e3 + 2.91e3i)25-s − 2.31e3i·27-s + 1.22e3·29-s + 7.81e3·31-s + ⋯
L(s)  = 1  − 0.321i·3-s + (0.825 + 0.564i)5-s − 0.118i·7-s + 0.896·9-s + 1.43·11-s − 0.996i·13-s + (0.181 − 0.265i)15-s − 1.68i·17-s − 1.53·19-s − 0.0382·21-s + 1.71i·23-s + (0.362 + 0.931i)25-s − 0.610i·27-s + 0.270·29-s + 1.46·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.892068521\)
\(L(\frac12)\) \(\approx\) \(2.892068521\)
\(L(\frac{7}{2})\) 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 \( 1 + (-46.1 - 31.5i)T \)
good3 \( 1 + 5.01iT - 243T^{2} \)
7 \( 1 + 15.3iT - 1.68e4T^{2} \)
11 \( 1 - 576.T + 1.61e5T^{2} \)
13 \( 1 + 607. iT - 3.71e5T^{2} \)
17 \( 1 + 2.01e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.42e3T + 2.47e6T^{2} \)
23 \( 1 - 4.35e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.22e3T + 2.05e7T^{2} \)
31 \( 1 - 7.81e3T + 2.86e7T^{2} \)
37 \( 1 + 3.80e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.47e3T + 1.15e8T^{2} \)
43 \( 1 + 1.47e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.04e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.09e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.11e4T + 7.14e8T^{2} \)
61 \( 1 - 4.48e3T + 8.44e8T^{2} \)
67 \( 1 + 4.54e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.03e4T + 1.80e9T^{2} \)
73 \( 1 + 3.89e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.32e4T + 3.07e9T^{2} \)
83 \( 1 - 3.47e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.78e4T + 5.58e9T^{2} \)
97 \( 1 + 1.51e5iT - 8.58e9T^{2} \)
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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

−10.60677092382503724725992370884, −9.778308118119952324675201712291, −9.043205674628183769516669431911, −7.60584534584576225590465168909, −6.81405931451217494905635042058, −6.00622440664324360942200081748, −4.67953464996570925169652670155, −3.35275553510667634442852487417, −2.02857732182842569358345912530, −0.881182527394539181340315279785, 1.19917497667526726619769752932, 2.11165092607625617049734811000, 4.11231327721539903976566575501, 4.54808063066175259983296838593, 6.33688142988958906809732131018, 6.53615585874263763716311016048, 8.442226150655136967581658181724, 8.947537346906880855286020414369, 10.01636672520421218772480296432, 10.60438011315190243203378072292

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