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Properties

Label 2-1920-80.3-c1-0-13
Degree $2$
Conductor $1920$
Sign $-0.323 - 0.946i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
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  + 3-s + (0.154 + 2.23i)5-s + (−1.27 + 1.27i)7-s + 9-s + (3.66 + 3.66i)11-s + 0.758i·13-s + (0.154 + 2.23i)15-s + (0.715 − 0.715i)17-s + (−0.203 − 0.203i)19-s + (−1.27 + 1.27i)21-s + (1.11 + 1.11i)23-s + (−4.95 + 0.689i)25-s + 27-s + (−6.80 + 6.80i)29-s − 6.28i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.0690 + 0.997i)5-s + (−0.483 + 0.483i)7-s + 0.333·9-s + (1.10 + 1.10i)11-s + 0.210i·13-s + (0.0398 + 0.575i)15-s + (0.173 − 0.173i)17-s + (−0.0466 − 0.0466i)19-s + (−0.279 + 0.279i)21-s + (0.231 + 0.231i)23-s + (−0.990 + 0.137i)25-s + 0.192·27-s + (−1.26 + 1.26i)29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.932257548\)
\(L(\frac12)\) \(\approx\) \(1.932257548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + (-0.154 - 2.23i)T \)
good7 \( 1 + (1.27 - 1.27i)T - 7iT^{2} \)
11 \( 1 + (-3.66 - 3.66i)T + 11iT^{2} \)
13 \( 1 - 0.758iT - 13T^{2} \)
17 \( 1 + (-0.715 + 0.715i)T - 17iT^{2} \)
19 \( 1 + (0.203 + 0.203i)T + 19iT^{2} \)
23 \( 1 + (-1.11 - 1.11i)T + 23iT^{2} \)
29 \( 1 + (6.80 - 6.80i)T - 29iT^{2} \)
31 \( 1 + 6.28iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 4.74iT - 43T^{2} \)
47 \( 1 + (-7.94 - 7.94i)T + 47iT^{2} \)
53 \( 1 + 5.01T + 53T^{2} \)
59 \( 1 + (4.46 - 4.46i)T - 59iT^{2} \)
61 \( 1 + (3.88 + 3.88i)T + 61iT^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 + 4.57T + 71T^{2} \)
73 \( 1 + (6.99 - 6.99i)T - 73iT^{2} \)
79 \( 1 - 4.49T + 79T^{2} \)
83 \( 1 - 17.9T + 83T^{2} \)
89 \( 1 + 1.53T + 89T^{2} \)
97 \( 1 + (5.91 - 5.91i)T - 97iT^{2} \)
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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

−9.244347609116021938402077792883, −9.035328500344960760526110741843, −7.60387858970233742014798671084, −7.19046584249738415561140060726, −6.42113396639101170821825039473, −5.56837710207924232378024248550, −4.27503091793891582287311093043, −3.54771364031191982391387111378, −2.59075504353168761910682618838, −1.69764091576135791387625046662, 0.65485211579826632260478191486, 1.73977042422886512446059371830, 3.21947309558619472722625067314, 3.87056637278571627709997542605, 4.76018496809064749857737409564, 5.87166709246027034141458051012, 6.49976444613709648934356031790, 7.60529879336704255187220805199, 8.261480795285188221600224741439, 9.059469627012318137959140633053

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