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Properties

Label 2-3840-3840.389-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.985 + 0.170i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.427 − 0.903i)2-s + (0.989 + 0.146i)3-s + (−0.634 + 0.773i)4-s + (−0.0490 + 0.998i)5-s + (−0.290 − 0.956i)6-s + (0.970 + 0.242i)8-s + (0.956 + 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.740 + 0.671i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (−0.262 − 1.31i)17-s + (−0.146 − 0.989i)18-s + (1.75 − 0.829i)19-s + (−0.740 − 0.671i)20-s + ⋯
L(s)  = 1  + (−0.427 − 0.903i)2-s + (0.989 + 0.146i)3-s + (−0.634 + 0.773i)4-s + (−0.0490 + 0.998i)5-s + (−0.290 − 0.956i)6-s + (0.970 + 0.242i)8-s + (0.956 + 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.740 + 0.671i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (−0.262 − 1.31i)17-s + (−0.146 − 0.989i)18-s + (1.75 − 0.829i)19-s + (−0.740 − 0.671i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.492182308\)
\(L(\frac12)\) \(\approx\) \(1.492182308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.427 + 0.903i)T \)
3 \( 1 + (-0.989 - 0.146i)T \)
5 \( 1 + (0.0490 - 0.998i)T \)
good7 \( 1 + (-0.555 - 0.831i)T^{2} \)
11 \( 1 + (-0.471 + 0.881i)T^{2} \)
13 \( 1 + (-0.0980 - 0.995i)T^{2} \)
17 \( 1 + (0.262 + 1.31i)T + (-0.923 + 0.382i)T^{2} \)
19 \( 1 + (-1.75 + 0.829i)T + (0.634 - 0.773i)T^{2} \)
23 \( 1 + (-1.45 - 1.19i)T + (0.195 + 0.980i)T^{2} \)
29 \( 1 + (0.881 - 0.471i)T^{2} \)
31 \( 1 + (1.42 + 0.591i)T + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.773 + 0.634i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (0.956 - 0.290i)T^{2} \)
47 \( 1 + (0.892 - 1.33i)T + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (0.229 - 0.914i)T + (-0.881 - 0.471i)T^{2} \)
59 \( 1 + (-0.0980 + 0.995i)T^{2} \)
61 \( 1 + (-1.37 + 1.02i)T + (0.290 - 0.956i)T^{2} \)
67 \( 1 + (0.290 - 0.956i)T^{2} \)
71 \( 1 + (0.831 - 0.555i)T^{2} \)
73 \( 1 + (-0.555 + 0.831i)T^{2} \)
79 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (1.80 + 0.644i)T + (0.773 + 0.634i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−9.032006801504224101468754155448, −7.82433210476271862175336124958, −7.38326206236523169526695557557, −6.96424716340641271627978095841, −5.40905665364961831203757525439, −4.63403609024698605100801686290, −3.54954306333793174737635763956, −3.05505631024397324996747412156, −2.46754975456900432460918888431, −1.26554226241394541081935663515, 1.08864270270663427965427931751, 1.91140482288064203294563572308, 3.42946937168223248303361666698, 4.14185280262043761094226559916, 5.09462479642737136930947110102, 5.64878359034460010161831742633, 6.79373128819924875777255702140, 7.27277716606423118113650504295, 8.216581098994858107101640515178, 8.507326111158054886977595552047

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