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Properties

Label 2-320-16.11-c2-0-2
Degree $2$
Conductor $320$
Sign $0.604 - 0.796i$
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  + (−3.58 − 3.58i)3-s + (−1.58 − 1.58i)5-s − 10.0·7-s + 16.7i·9-s + (0.304 − 0.304i)11-s + (6.87 − 6.87i)13-s + 11.3i·15-s − 25.8·17-s + (10.2 + 10.2i)19-s + (35.8 + 35.8i)21-s + 33.7·23-s + 5.00i·25-s + (27.6 − 27.6i)27-s + (23.1 − 23.1i)29-s + 7.31i·31-s + ⋯
L(s)  = 1  + (−1.19 − 1.19i)3-s + (−0.316 − 0.316i)5-s − 1.42·7-s + 1.85i·9-s + (0.0276 − 0.0276i)11-s + (0.528 − 0.528i)13-s + 0.755i·15-s − 1.51·17-s + (0.539 + 0.539i)19-s + (1.70 + 1.70i)21-s + 1.46·23-s + 0.200i·25-s + (1.02 − 1.02i)27-s + (0.798 − 0.798i)29-s + 0.236i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.604 - 0.796i$
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: \((2,\ 320,\ (\ :1),\ 0.604 - 0.796i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.278652 + 0.138260i\)
\(L(\frac12)\) \(\approx\) \(0.278652 + 0.138260i\)
\(L(2)\) 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 \( 1 + (3.58 + 3.58i)T + 9iT^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + (-0.304 + 0.304i)T - 121iT^{2} \)
13 \( 1 + (-6.87 + 6.87i)T - 169iT^{2} \)
17 \( 1 + 25.8T + 289T^{2} \)
19 \( 1 + (-10.2 - 10.2i)T + 361iT^{2} \)
23 \( 1 - 33.7T + 529T^{2} \)
29 \( 1 + (-23.1 + 23.1i)T - 841iT^{2} \)
31 \( 1 - 7.31iT - 961T^{2} \)
37 \( 1 + (-6.85 - 6.85i)T + 1.36e3iT^{2} \)
41 \( 1 - 63.7iT - 1.68e3T^{2} \)
43 \( 1 + (15.0 - 15.0i)T - 1.84e3iT^{2} \)
47 \( 1 - 41.8iT - 2.20e3T^{2} \)
53 \( 1 + (36.5 + 36.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (58.1 - 58.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (26.2 - 26.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-4.48 - 4.48i)T + 4.48e3iT^{2} \)
71 \( 1 + 11.5T + 5.04e3T^{2} \)
73 \( 1 - 101. iT - 5.32e3T^{2} \)
79 \( 1 + 73.1iT - 6.24e3T^{2} \)
83 \( 1 + (39.9 + 39.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 37.2iT - 7.92e3T^{2} \)
97 \( 1 + 114.T + 9.40e3T^{2} \)
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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

−11.60617767244312166221049204291, −10.94159531666061695345629354035, −9.794682255110033166966723651207, −8.601963621835682308820833877340, −7.43851068329882812664833541170, −6.53072064892871709699197777444, −6.00322194240547156288420578774, −4.69303493701383283675106719899, −2.98263845965888443216015111266, −1.07639281982568945011452579740, 0.20770320036981087950585025037, 3.13738415345441556515003878996, 4.16797974857812670782790571009, 5.20457651417641428820885122578, 6.44408576694858847912984836158, 6.89342856350970120509095671225, 8.953979025773997942004975213519, 9.460671603970168797131661597072, 10.61184606679628233738140494554, 11.01924360919743551505224525844

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