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Properties

Label 2-320-80.53-c2-0-20
Degree $2$
Conductor $320$
Sign $0.0988 + 0.995i$
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  + 4.95·3-s + (−3.04 − 3.96i)5-s + (−7.61 − 7.61i)7-s + 15.5·9-s + (−5.22 − 5.22i)11-s + 4.80·13-s + (−15.0 − 19.6i)15-s + (16.0 − 16.0i)17-s + (−6.55 − 6.55i)19-s + (−37.6 − 37.6i)21-s + (5.38 − 5.38i)23-s + (−6.47 + 24.1i)25-s + 32.2·27-s + (22.0 + 22.0i)29-s + 9.92·31-s + ⋯
L(s)  = 1  + 1.65·3-s + (−0.608 − 0.793i)5-s + (−1.08 − 1.08i)7-s + 1.72·9-s + (−0.475 − 0.475i)11-s + 0.369·13-s + (−1.00 − 1.30i)15-s + (0.942 − 0.942i)17-s + (−0.345 − 0.345i)19-s + (−1.79 − 1.79i)21-s + (0.234 − 0.234i)23-s + (−0.258 + 0.965i)25-s + 1.19·27-s + (0.759 + 0.759i)29-s + 0.320·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0988 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0988 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.0988 + 0.995i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.0988 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61257 - 1.46027i\)
\(L(\frac12)\) \(\approx\) \(1.61257 - 1.46027i\)
\(L(2)\) 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 \( 1 + (3.04 + 3.96i)T \)
good3 \( 1 - 4.95T + 9T^{2} \)
7 \( 1 + (7.61 + 7.61i)T + 49iT^{2} \)
11 \( 1 + (5.22 + 5.22i)T + 121iT^{2} \)
13 \( 1 - 4.80T + 169T^{2} \)
17 \( 1 + (-16.0 + 16.0i)T - 289iT^{2} \)
19 \( 1 + (6.55 + 6.55i)T + 361iT^{2} \)
23 \( 1 + (-5.38 + 5.38i)T - 529iT^{2} \)
29 \( 1 + (-22.0 - 22.0i)T + 841iT^{2} \)
31 \( 1 - 9.92T + 961T^{2} \)
37 \( 1 - 32.0T + 1.36e3T^{2} \)
41 \( 1 - 44.7iT - 1.68e3T^{2} \)
43 \( 1 + 1.64iT - 1.84e3T^{2} \)
47 \( 1 + (44.0 - 44.0i)T - 2.20e3iT^{2} \)
53 \( 1 - 2.84iT - 2.80e3T^{2} \)
59 \( 1 + (-19.9 + 19.9i)T - 3.48e3iT^{2} \)
61 \( 1 + (-62.5 + 62.5i)T - 3.72e3iT^{2} \)
67 \( 1 + 88.2iT - 4.48e3T^{2} \)
71 \( 1 + 36.9iT - 5.04e3T^{2} \)
73 \( 1 + (-22.6 + 22.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 125. iT - 6.24e3T^{2} \)
83 \( 1 - 105.T + 6.88e3T^{2} \)
89 \( 1 + 66.0T + 7.92e3T^{2} \)
97 \( 1 + (16.0 - 16.0i)T - 9.40e3iT^{2} \)
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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

−11.08454004174810925534520415911, −9.901838645972273903444238917247, −9.300993425025318138019044494377, −8.271248969170862799427796338815, −7.69290441675335344889156325369, −6.64028807466192914415684856364, −4.78962815756778177481059785181, −3.64096746841282752790716099529, −2.95430701597286063662927379700, −0.878909188932880458301016124313, 2.29602464631386724153343146934, 3.12578924696424404960393212337, 3.98008375234038279282035686700, 5.91020698615206243770288980442, 7.02330886863460798023795707673, 8.053069226197563577413523498905, 8.640002105931425321052466715808, 9.765311336069153380171209431452, 10.30734509169782064474173155736, 11.82569938366680334492692556979

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