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Properties

Label 2-320-20.19-c2-0-3
Degree $2$
Conductor $320$
Sign $-0.509 - 0.860i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.75·3-s + (−4.30 + 2.54i)5-s − 3.84·7-s − 1.41·9-s + 6.19i·11-s + 16.1i·13-s + (−11.8 + 7.01i)15-s − 5.20i·17-s + 36.2i·19-s − 10.6·21-s − 22.0·23-s + (12.0 − 21.9i)25-s − 28.6·27-s + 20.0·29-s + 26.4i·31-s + ⋯
L(s)  = 1  + 0.918·3-s + (−0.860 + 0.509i)5-s − 0.549·7-s − 0.157·9-s + 0.562i·11-s + 1.23i·13-s + (−0.789 + 0.467i)15-s − 0.306i·17-s + 1.90i·19-s − 0.504·21-s − 0.958·23-s + (0.480 − 0.876i)25-s − 1.06·27-s + 0.690·29-s + 0.852i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.569309 + 0.998800i\)
\(L(\frac12)\) \(\approx\) \(0.569309 + 0.998800i\)
\(L(2)\) 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 \( 1 + (4.30 - 2.54i)T \)
good3 \( 1 - 2.75T + 9T^{2} \)
7 \( 1 + 3.84T + 49T^{2} \)
11 \( 1 - 6.19iT - 121T^{2} \)
13 \( 1 - 16.1iT - 169T^{2} \)
17 \( 1 + 5.20iT - 289T^{2} \)
19 \( 1 - 36.2iT - 361T^{2} \)
23 \( 1 + 22.0T + 529T^{2} \)
29 \( 1 - 20.0T + 841T^{2} \)
31 \( 1 - 26.4iT - 961T^{2} \)
37 \( 1 + 69.3iT - 1.36e3T^{2} \)
41 \( 1 - 11.6T + 1.68e3T^{2} \)
43 \( 1 + 25.8T + 1.84e3T^{2} \)
47 \( 1 - 66.1T + 2.20e3T^{2} \)
53 \( 1 - 39.5iT - 2.80e3T^{2} \)
59 \( 1 - 27.7iT - 3.48e3T^{2} \)
61 \( 1 - 54.1T + 3.72e3T^{2} \)
67 \( 1 + 107.T + 4.48e3T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 37.4iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 126.T + 6.88e3T^{2} \)
89 \( 1 - 133.T + 7.92e3T^{2} \)
97 \( 1 - 6.40iT - 9.40e3T^{2} \)
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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

−11.94597275846186719167373695704, −10.72283289179252834994049702905, −9.767933290161720482186019582461, −8.855197806449527835423536015335, −7.927834244479482774302270786527, −7.12567616712491269998000023310, −6.00523295558657510294123113388, −4.22750422705702766620509826304, −3.44365412428792927456027975228, −2.15288949693950947600483807677, 0.46868823325521407782078802965, 2.74772341855579465579377837742, 3.57897855638254437751457411867, 4.90908254807366358396367347867, 6.23196446552555725300768410311, 7.58447396936173104882702383564, 8.330812386426172644270936008568, 8.967176434086083931896883545396, 10.04948548997752523545772696659, 11.19545579304025996830698510031

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