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Properties

Label 2-3840-8.5-c1-0-35
Degree $2$
Conductor $3840$
Sign $0.707 + 0.707i$
Analytic cond. $30.6625$
Root an. cond. $5.53737$
Motivic weight $1$
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 − 4i·11-s − 2i·13-s + 15-s − 2·17-s + 8i·19-s + 4·23-s − 25-s − i·27-s + 6i·29-s + 4·33-s + 2i·37-s + 2·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s − 0.333·9-s − 1.20i·11-s − 0.554i·13-s + 0.258·15-s − 0.485·17-s + 1.83i·19-s + 0.834·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s + 0.696·33-s + 0.328i·37-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1921, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.589861658\)
\(L(\frac12)\) \(\approx\) \(1.589861658\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.319125142455782124072716279994, −8.002363420704153567279291899771, −6.85838879498153382910980243468, −5.96278195874314239303295123858, −5.43742737680556945262430087325, −4.65064048627579760159166614043, −3.63134600332045456906158320176, −3.16054607192989109082607551240, −1.79947100943392382824594943396, −0.54422233506717417579270614218, 1.00478179388458173121284552246, 2.32824668932091512799559347122, 2.70368152227799927533369374741, 4.15657122838561286427890235461, 4.66803738592479078461342966742, 5.73294573375495546384283384345, 6.54979762922825881444686422125, 7.24508239503512494838217655427, 7.46042777954692204712799623731, 8.697402441538386350238848742775

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