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Properties

Label 2-3840-3840.1709-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.817 + 0.575i$
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  + (0.0490 − 0.998i)2-s + (0.514 + 0.857i)3-s + (−0.995 − 0.0980i)4-s + (0.336 − 0.941i)5-s + (0.881 − 0.471i)6-s + (−0.146 + 0.989i)8-s + (−0.471 + 0.881i)9-s + (−0.923 − 0.382i)10-s + (−0.427 − 0.903i)12-s + (0.980 − 0.195i)15-s + (0.980 + 0.195i)16-s + (1.77 + 0.352i)17-s + (0.857 + 0.514i)18-s + (0.293 + 0.0143i)19-s + (−0.427 + 0.903i)20-s + ⋯
L(s)  = 1  + (0.0490 − 0.998i)2-s + (0.514 + 0.857i)3-s + (−0.995 − 0.0980i)4-s + (0.336 − 0.941i)5-s + (0.881 − 0.471i)6-s + (−0.146 + 0.989i)8-s + (−0.471 + 0.881i)9-s + (−0.923 − 0.382i)10-s + (−0.427 − 0.903i)12-s + (0.980 − 0.195i)15-s + (0.980 + 0.195i)16-s + (1.77 + 0.352i)17-s + (0.857 + 0.514i)18-s + (0.293 + 0.0143i)19-s + (−0.427 + 0.903i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547796171\)
\(L(\frac12)\) \(\approx\) \(1.547796171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0490 + 0.998i)T \)
3 \( 1 + (-0.514 - 0.857i)T \)
5 \( 1 + (-0.336 + 0.941i)T \)
good7 \( 1 + (-0.831 - 0.555i)T^{2} \)
11 \( 1 + (-0.290 + 0.956i)T^{2} \)
13 \( 1 + (0.634 + 0.773i)T^{2} \)
17 \( 1 + (-1.77 - 0.352i)T + (0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.293 - 0.0143i)T + (0.995 + 0.0980i)T^{2} \)
23 \( 1 + (0.145 - 1.47i)T + (-0.980 - 0.195i)T^{2} \)
29 \( 1 + (-0.956 + 0.290i)T^{2} \)
31 \( 1 + (-0.181 + 0.0750i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.0980 - 0.995i)T^{2} \)
41 \( 1 + (-0.195 + 0.980i)T^{2} \)
43 \( 1 + (-0.471 - 0.881i)T^{2} \)
47 \( 1 + (-0.404 + 0.269i)T + (0.382 - 0.923i)T^{2} \)
53 \( 1 + (-0.574 - 0.0851i)T + (0.956 + 0.290i)T^{2} \)
59 \( 1 + (0.634 - 0.773i)T^{2} \)
61 \( 1 + (-0.390 + 1.55i)T + (-0.881 - 0.471i)T^{2} \)
67 \( 1 + (-0.881 - 0.471i)T^{2} \)
71 \( 1 + (-0.555 + 0.831i)T^{2} \)
73 \( 1 + (-0.831 + 0.555i)T^{2} \)
79 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.698 - 0.633i)T + (0.0980 - 0.995i)T^{2} \)
89 \( 1 + (0.980 - 0.195i)T^{2} \)
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

−8.837703990590793528774063406132, −8.108713225245269429552442709807, −7.58861360629436449747737538757, −5.81213523759663097261622829203, −5.42538906509820984018495801499, −4.68437751640060550352855704386, −3.80029463018264226988793104407, −3.26770488388349878007206861585, −2.14631313322890035633347390395, −1.18269664870809742316742353317, 1.03271788724943338728817853611, 2.48119059993139024404462850774, 3.24519577524614751878416059524, 4.08986063915269881783256035517, 5.39449431734496505588286478940, 5.91113326906677415115268811953, 6.73010913645476971330694824505, 7.19030813674186631523910751942, 7.85475319896357741422668171968, 8.457126180747608084529700383251

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