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Properties

Label 2-3840-3840.749-c0-0-1
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  + (0.740 − 0.671i)2-s + (0.242 − 0.970i)3-s + (0.0980 − 0.995i)4-s + (−0.427 − 0.903i)5-s + (−0.471 − 0.881i)6-s + (−0.595 − 0.803i)8-s + (−0.881 − 0.471i)9-s + (−0.923 − 0.382i)10-s + (−0.941 − 0.336i)12-s + (−0.980 + 0.195i)15-s + (−0.980 − 0.195i)16-s + (−0.660 − 0.131i)17-s + (−0.970 + 0.242i)18-s + (0.800 + 0.882i)19-s + (−0.941 + 0.336i)20-s + ⋯
L(s)  = 1  + (0.740 − 0.671i)2-s + (0.242 − 0.970i)3-s + (0.0980 − 0.995i)4-s + (−0.427 − 0.903i)5-s + (−0.471 − 0.881i)6-s + (−0.595 − 0.803i)8-s + (−0.881 − 0.471i)9-s + (−0.923 − 0.382i)10-s + (−0.941 − 0.336i)12-s + (−0.980 + 0.195i)15-s + (−0.980 − 0.195i)16-s + (−0.660 − 0.131i)17-s + (−0.970 + 0.242i)18-s + (0.800 + 0.882i)19-s + (−0.941 + 0.336i)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: $\chi_{3840} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ -0.817 - 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.499740948\)
\(L(\frac12)\) \(\approx\) \(1.499740948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.740 + 0.671i)T \)
3 \( 1 + (-0.242 + 0.970i)T \)
5 \( 1 + (0.427 + 0.903i)T \)
good7 \( 1 + (0.831 + 0.555i)T^{2} \)
11 \( 1 + (0.956 + 0.290i)T^{2} \)
13 \( 1 + (-0.773 + 0.634i)T^{2} \)
17 \( 1 + (0.660 + 0.131i)T + (0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.800 - 0.882i)T + (-0.0980 + 0.995i)T^{2} \)
23 \( 1 + (1.98 + 0.195i)T + (0.980 + 0.195i)T^{2} \)
29 \( 1 + (-0.290 - 0.956i)T^{2} \)
31 \( 1 + (-1.83 + 0.761i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.995 + 0.0980i)T^{2} \)
41 \( 1 + (0.195 - 0.980i)T^{2} \)
43 \( 1 + (-0.881 + 0.471i)T^{2} \)
47 \( 1 + (-1.42 + 0.953i)T + (0.382 - 0.923i)T^{2} \)
53 \( 1 + (-1.53 + 1.14i)T + (0.290 - 0.956i)T^{2} \)
59 \( 1 + (-0.773 - 0.634i)T^{2} \)
61 \( 1 + (-0.251 + 0.150i)T + (0.471 - 0.881i)T^{2} \)
67 \( 1 + (0.471 - 0.881i)T^{2} \)
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 \( 1 + (1.76 + 0.0865i)T + (0.995 + 0.0980i)T^{2} \)
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

−8.223487121211983987775715134854, −7.55984020140046078260187430969, −6.58474665371451284427558295096, −5.91034036726663136366484227229, −5.23088321762676773237149273784, −4.22044131144645058630579309815, −3.63664464316761152489115062227, −2.47920494829858324264441080552, −1.72674150906500756238867351953, −0.61379929729496158668807072657, 2.48895152467001823152924087665, 2.96543449505625056625435927575, 4.02965929655628254713737820968, 4.33437704959725202696775665623, 5.35447425707164748664438317973, 6.10997803179538700297651363829, 6.79320312080301484870127084093, 7.65650135704676878936244980319, 8.200985105104732302460399340381, 8.951788964001756294278194647280

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