Properties

Label 2-320-16.11-c2-0-8
Degree $2$
Conductor $320$
Sign $0.999 + 0.0135i$
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  + (1.45 + 1.45i)3-s + (−1.58 − 1.58i)5-s + 11.3·7-s − 4.75i·9-s + (1.50 − 1.50i)11-s + (−0.454 + 0.454i)13-s − 4.60i·15-s − 1.99·17-s + (−5.07 − 5.07i)19-s + (16.6 + 16.6i)21-s + 41.9·23-s + 5.00i·25-s + (20.0 − 20.0i)27-s + (7.01 − 7.01i)29-s + 33.3i·31-s + ⋯
L(s)  = 1  + (0.485 + 0.485i)3-s + (−0.316 − 0.316i)5-s + 1.62·7-s − 0.527i·9-s + (0.137 − 0.137i)11-s + (−0.0349 + 0.0349i)13-s − 0.307i·15-s − 0.117·17-s + (−0.267 − 0.267i)19-s + (0.790 + 0.790i)21-s + 1.82·23-s + 0.200i·25-s + (0.742 − 0.742i)27-s + (0.242 − 0.242i)29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.999 + 0.0135i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.999 + 0.0135i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.18578 - 0.0148196i\)
\(L(\frac12)\) \(\approx\) \(2.18578 - 0.0148196i\)
\(L(2)\) 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 \( 1 + (1.58 + 1.58i)T \)
good3 \( 1 + (-1.45 - 1.45i)T + 9iT^{2} \)
7 \( 1 - 11.3T + 49T^{2} \)
11 \( 1 + (-1.50 + 1.50i)T - 121iT^{2} \)
13 \( 1 + (0.454 - 0.454i)T - 169iT^{2} \)
17 \( 1 + 1.99T + 289T^{2} \)
19 \( 1 + (5.07 + 5.07i)T + 361iT^{2} \)
23 \( 1 - 41.9T + 529T^{2} \)
29 \( 1 + (-7.01 + 7.01i)T - 841iT^{2} \)
31 \( 1 - 33.3iT - 961T^{2} \)
37 \( 1 + (-44.5 - 44.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 51.6iT - 1.68e3T^{2} \)
43 \( 1 + (37.7 - 37.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 16.2iT - 2.20e3T^{2} \)
53 \( 1 + (67.7 + 67.7i)T + 2.80e3iT^{2} \)
59 \( 1 + (34.2 - 34.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (67.1 - 67.1i)T - 3.72e3iT^{2} \)
67 \( 1 + (-9.87 - 9.87i)T + 4.48e3iT^{2} \)
71 \( 1 - 74.7T + 5.04e3T^{2} \)
73 \( 1 + 101. iT - 5.32e3T^{2} \)
79 \( 1 + 63.6iT - 6.24e3T^{2} \)
83 \( 1 + (57.1 + 57.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 33.6iT - 7.92e3T^{2} \)
97 \( 1 + 98.2T + 9.40e3T^{2} \)
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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

−11.37623508453809087888490098478, −10.60441582661611514329933790308, −9.311722583344569929944216591839, −8.631671016077727805525121776053, −7.85747868156416614634777896927, −6.63195963270281358164190690510, −5.05796080101571466361487022392, −4.39725161317671726754263277153, −3.03056227681730985803543676742, −1.26633624857277468070774660844, 1.47788157618438456777553282670, 2.69174205985718114754917519223, 4.34990810471309154292343928712, 5.28528739573401216603406880551, 6.83928113722855341010049610195, 7.79829705564383203865927445967, 8.257353620446253433176949165310, 9.390955472291715205554716330049, 11.00436849582046873041276070844, 11.05949848214269696895048268447

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