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Properties

Label 2-3840-3840.509-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.932 - 0.359i$
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  + (−0.970 − 0.242i)2-s + (−0.903 − 0.427i)3-s + (0.881 + 0.471i)4-s + (−0.146 + 0.989i)5-s + (0.773 + 0.634i)6-s + (−0.740 − 0.671i)8-s + (0.634 + 0.773i)9-s + (0.382 − 0.923i)10-s + (−0.595 − 0.803i)12-s + (0.555 − 0.831i)15-s + (0.555 + 0.831i)16-s + (0.892 + 1.33i)17-s + (−0.427 − 0.903i)18-s + (0.360 − 1.43i)19-s + (−0.595 + 0.803i)20-s + ⋯
L(s)  = 1  + (−0.970 − 0.242i)2-s + (−0.903 − 0.427i)3-s + (0.881 + 0.471i)4-s + (−0.146 + 0.989i)5-s + (0.773 + 0.634i)6-s + (−0.740 − 0.671i)8-s + (0.634 + 0.773i)9-s + (0.382 − 0.923i)10-s + (−0.595 − 0.803i)12-s + (0.555 − 0.831i)15-s + (0.555 + 0.831i)16-s + (0.892 + 1.33i)17-s + (−0.427 − 0.903i)18-s + (0.360 − 1.43i)19-s + (−0.595 + 0.803i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.932 - 0.359i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.932 - 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5954541786\)
\(L(\frac12)\) \(\approx\) \(0.5954541786\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.970 + 0.242i)T \)
3 \( 1 + (0.903 + 0.427i)T \)
5 \( 1 + (0.146 - 0.989i)T \)
good7 \( 1 + (0.980 - 0.195i)T^{2} \)
11 \( 1 + (0.995 - 0.0980i)T^{2} \)
13 \( 1 + (0.290 + 0.956i)T^{2} \)
17 \( 1 + (-0.892 - 1.33i)T + (-0.382 + 0.923i)T^{2} \)
19 \( 1 + (-0.360 + 1.43i)T + (-0.881 - 0.471i)T^{2} \)
23 \( 1 + (-0.484 + 0.906i)T + (-0.555 - 0.831i)T^{2} \)
29 \( 1 + (0.0980 - 0.995i)T^{2} \)
31 \( 1 + (-0.360 - 0.871i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.471 + 0.881i)T^{2} \)
41 \( 1 + (-0.831 + 0.555i)T^{2} \)
43 \( 1 + (0.634 - 0.773i)T^{2} \)
47 \( 1 + (-0.660 - 0.131i)T + (0.923 + 0.382i)T^{2} \)
53 \( 1 + (-1.33 + 1.47i)T + (-0.0980 - 0.995i)T^{2} \)
59 \( 1 + (0.290 - 0.956i)T^{2} \)
61 \( 1 + (-0.0330 - 0.0923i)T + (-0.773 + 0.634i)T^{2} \)
67 \( 1 + (-0.773 + 0.634i)T^{2} \)
71 \( 1 + (-0.195 - 0.980i)T^{2} \)
73 \( 1 + (0.980 + 0.195i)T^{2} \)
79 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.652 - 1.08i)T + (-0.471 + 0.881i)T^{2} \)
89 \( 1 + (0.555 - 0.831i)T^{2} \)
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

−8.561037229028398232385713934830, −7.958242309914970265964721994439, −7.12959550006118144582397121317, −6.72466417335512987149585895670, −6.07935680786805162833405894701, −5.14195574587931284253323932362, −3.94462996706192801310761436186, −2.97314680291744473339681181635, −2.10550766929571115774780363179, −0.929294312456816746263715405994, 0.73349664924703096156751664879, 1.61139566048591198270924497536, 3.13947815792142794317507523997, 4.16693330708556542763373103520, 5.23500412667506727751892886324, 5.56470583801270104705308246292, 6.35042219287142879876835842681, 7.46346524765139971074692214694, 7.72087488733204717237401392499, 8.783925304220586176240962838689

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