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Properties

Label 2-3840-15.14-c0-0-0
Degree $2$
Conductor $3840$
Sign $-1$
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  + i·3-s + i·5-s − 9-s − 15-s − 2·23-s − 25-s − i·27-s + 2i·29-s + 2i·43-s − i·45-s − 2·47-s + 49-s − 2i·67-s − 2i·69-s − i·75-s + ⋯
L(s)  = 1  + i·3-s + i·5-s − 9-s − 15-s − 2·23-s − 25-s − i·27-s + 2i·29-s + 2i·43-s − i·45-s − 2·47-s + 49-s − 2i·67-s − 2i·69-s − i·75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (3329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7884684498\)
\(L(\frac12)\) \(\approx\) \(0.7884684498\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 - iT \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + 2T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.216061643017827151685357204053, −8.242222367061072285081183185936, −7.72526127635164750337061291731, −6.64609269182756929758085144868, −6.12027935400059718055719539966, −5.25569824908587430681230772475, −4.41608918204204362273344335240, −3.56136538980186347254417880380, −2.98583776041186282921870236072, −1.89936579663403351428624598486, 0.41810476472220982023342676168, 1.70622497737331762722787107917, 2.38858856766795997831811408380, 3.72158536537594341473587194973, 4.47492287836252459085782255554, 5.59367011314544884610359236579, 5.93761356620416056139961672163, 6.87024597621971873922727212485, 7.72450357588659115424928528501, 8.237249478880239461388427468817

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