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Properties

Label 2-1920-120.59-c1-0-22
Degree $2$
Conductor $1920$
Sign $-0.999 + 0.0430i$
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  + (0.618 + 1.61i)3-s + (1 + 2i)5-s − 1.23·7-s + (−2.23 + 2.00i)9-s + 4i·11-s − 2.47·13-s + (−2.61 + 2.85i)15-s − 2.47·17-s + 6.47·19-s + (−0.763 − 2.00i)21-s − 3.23i·23-s + (−3 + 4i)25-s + (−4.61 − 2.38i)27-s + 4.47·29-s + 6.47i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + (0.447 + 0.894i)5-s − 0.467·7-s + (−0.745 + 0.666i)9-s + 1.20i·11-s − 0.685·13-s + (−0.675 + 0.736i)15-s − 0.599·17-s + 1.48·19-s + (−0.166 − 0.436i)21-s − 0.674i·23-s + (−0.600 + 0.800i)25-s + (−0.888 − 0.458i)27-s + 0.830·29-s + 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.999 + 0.0430i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.999 + 0.0430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358483870\)
\(L(\frac12)\) \(\approx\) \(1.358483870\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 - 1.61i)T \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 6.47iT - 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 - 4.76iT - 43T^{2} \)
47 \( 1 + 4.76iT - 47T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 - 8.94iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 - 1.70iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 1.52iT - 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 - 4.94iT - 97T^{2} \)
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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

−9.734746229175175725729498634966, −9.085158616053260992401047126222, −8.086676484671409783052637775503, −7.07449960426598488745347540577, −6.65110360546312288973967458159, −5.34683142609885895839023787148, −4.81574271665069569587958182192, −3.65496844501323009813583875261, −2.89141154897237030512165251806, −2.02575835566290647547803316759, 0.44934588871415829609363851441, 1.53909520135487834531496044688, 2.72125564515835117949195986429, 3.55012628166787667464948530040, 4.90236504855492429881050590042, 5.73954561477660369910692330548, 6.35293877173931294965281898595, 7.34819216720719436382419859003, 8.017148268925577103842738485859, 8.865449977057221731687648201050

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