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Properties

Label 2-1920-80.69-c1-0-22
Degree $2$
Conductor $1920$
Sign $0.852 - 0.523i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.249730850\)
\(L(\frac12)\) \(\approx\) \(2.249730850\)
\(L(\frac{3}{2})\) 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

−8.784607794849323832245904224330, −8.514755199377903522396219015068, −8.127054539285063257043795366713, −7.11059699240665354126806138284, −5.87782056063720759050834240954, −5.07556789674647750880251696320, −4.17221012640910795012670596516, −3.84963197011660027724076057403, −2.12640524804751892902466553140, −1.26318975157236671304314734293, 0.921803525612720479522445919609, 2.24180724572735592037943455543, 3.06180153528939367530584867536, 4.20766444877858636687021479720, 4.97169328235462973928026088609, 6.00310143737852313672734942242, 7.05895356712665803629550244278, 7.60136165070905892952746070946, 8.359579172797591124721102382493, 8.678394403546457559670307870812

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