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Properties

Label 2-1920-80.29-c1-0-12
Degree $2$
Conductor $1920$
Sign $0.523 - 0.852i$
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  + (−0.707 + 0.707i)3-s + (−1.38 + 1.75i)5-s − 4.66·7-s − 1.00i·9-s + (1.23 − 1.23i)11-s + (4.12 − 4.12i)13-s + (−0.258 − 2.22i)15-s − 3.20i·17-s + (3.73 + 3.73i)19-s + (3.29 − 3.29i)21-s − 0.714·23-s + (−1.14 − 4.86i)25-s + (0.707 + 0.707i)27-s + (1.24 + 1.24i)29-s − 3.84·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.620 + 0.784i)5-s − 1.76·7-s − 0.333i·9-s + (0.372 − 0.372i)11-s + (1.14 − 1.14i)13-s + (−0.0666 − 0.573i)15-s − 0.778i·17-s + (0.857 + 0.857i)19-s + (0.719 − 0.719i)21-s − 0.149·23-s + (−0.229 − 0.973i)25-s + (0.136 + 0.136i)27-s + (0.230 + 0.230i)29-s − 0.690·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.523 - 0.852i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.523 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9228684610\)
\(L(\frac12)\) \(\approx\) \(0.9228684610\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.38 - 1.75i)T \)
good7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 + (-1.23 + 1.23i)T - 11iT^{2} \)
13 \( 1 + (-4.12 + 4.12i)T - 13iT^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 + (-3.73 - 3.73i)T + 19iT^{2} \)
23 \( 1 + 0.714T + 23T^{2} \)
29 \( 1 + (-1.24 - 1.24i)T + 29iT^{2} \)
31 \( 1 + 3.84T + 31T^{2} \)
37 \( 1 + (2.33 + 2.33i)T + 37iT^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 + (1.31 + 1.31i)T + 43iT^{2} \)
47 \( 1 - 1.18iT - 47T^{2} \)
53 \( 1 + (-9.35 - 9.35i)T + 53iT^{2} \)
59 \( 1 + (6.22 - 6.22i)T - 59iT^{2} \)
61 \( 1 + (-4.44 - 4.44i)T + 61iT^{2} \)
67 \( 1 + (6.37 - 6.37i)T - 67iT^{2} \)
71 \( 1 - 6.23iT - 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (-4.88 + 4.88i)T - 83iT^{2} \)
89 \( 1 + 2.20iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−9.432158996661967910498400052345, −8.669095259041177218116240396504, −7.60451691052896539679590530202, −6.90738813808176725398710086591, −6.04474695862752655009637170443, −5.65049865433250483596459052504, −4.12189378115910545135487923050, −3.34817903693248909768316429254, −2.98689388717750325258613404418, −0.76717643626469972260301087216, 0.55755371089793093628035907441, 1.84069514475751146004864602914, 3.41815585484513051855446889131, 3.95710849327338844594112049536, 5.06149860899351040116928246516, 6.08045007209220200647757418998, 6.67455826429578467544308834925, 7.30490509414523072999438344229, 8.415915040968643072792986309536, 9.126201579807886599794729458842

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