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Properties

Label 2-320-40.27-c1-0-7
Degree $2$
Conductor $320$
Sign $0.852 + 0.522i$
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  + (0.456 − 0.456i)3-s + (2.18 − 0.456i)5-s + (2.79 − 2.79i)7-s + 2.58i·9-s − 4.37·11-s + (−1.73 − 1.73i)13-s + (0.791 − 1.20i)15-s + (3 + 3i)17-s − 3.46i·19-s − 2.55i·21-s + (0.791 + 0.791i)23-s + (4.58 − 1.99i)25-s + (2.55 + 2.55i)27-s + 5.29·29-s − 1.58i·31-s + ⋯
L(s)  = 1  + (0.263 − 0.263i)3-s + (0.978 − 0.204i)5-s + (1.05 − 1.05i)7-s + 0.860i·9-s − 1.31·11-s + (−0.480 − 0.480i)13-s + (0.204 − 0.312i)15-s + (0.727 + 0.727i)17-s − 0.794i·19-s − 0.556i·21-s + (0.164 + 0.164i)23-s + (0.916 − 0.399i)25-s + (0.490 + 0.490i)27-s + 0.982·29-s − 0.284i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63729 - 0.461662i\)
\(L(\frac12)\) \(\approx\) \(1.63729 - 0.461662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.18 + 0.456i)T \)
good3 \( 1 + (-0.456 + 0.456i)T - 3iT^{2} \)
7 \( 1 + (-2.79 + 2.79i)T - 7iT^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-0.791 - 0.791i)T + 23iT^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 1.58iT - 31T^{2} \)
37 \( 1 + (5.19 - 5.19i)T - 37iT^{2} \)
41 \( 1 + 7.58T + 41T^{2} \)
43 \( 1 + (8.29 - 8.29i)T - 43iT^{2} \)
47 \( 1 + (0.791 - 0.791i)T - 47iT^{2} \)
53 \( 1 + (2.64 + 2.64i)T + 53iT^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 9.66iT - 61T^{2} \)
67 \( 1 + (-8.29 - 8.29i)T + 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (-0.582 + 0.582i)T - 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (-4.83 + 4.83i)T - 83iT^{2} \)
89 \( 1 + 3.16iT - 89T^{2} \)
97 \( 1 + (-0.582 - 0.582i)T + 97iT^{2} \)
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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

−11.37282595454449196833953317961, −10.36254226096584407145296461251, −10.10265259806200670970660982133, −8.386293588671690289914134759928, −7.918584000673338128116486956810, −6.86996572792364103692108380625, −5.29585085309259754788388190565, −4.79633566736593300562678079410, −2.81956669829911059749810591717, −1.51776662830399080839565989131, 1.97046487876769149429060984042, 3.09087052806274083177897050892, 4.96393915148140038987392594617, 5.55949156749524866516973330256, 6.82925488896532225918105148259, 8.123563353287430431527244767704, 8.939160980827677868507278916012, 9.858142998421329586247097934271, 10.59224965301357593165731165309, 11.89747018291850393775597208662

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