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Properties

Label 2-3840-15.14-c0-0-1
Degree $2$
Conductor $3840$
Sign $-i$
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  − 3-s − i·5-s + 2i·7-s + 9-s + i·15-s − 2i·21-s − 25-s − 27-s + 2i·29-s + 2·35-s − i·45-s − 3·49-s + 2i·63-s + 75-s + 81-s + ⋯
L(s)  = 1  − 3-s − i·5-s + 2i·7-s + 9-s + i·15-s − 2i·21-s − 25-s − 27-s + 2i·29-s + 2·35-s − i·45-s − 3·49-s + 2i·63-s + 75-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-i$
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: \((2,\ 3840,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7053144522\)
\(L(\frac12)\) \(\approx\) \(0.7053144522\)
\(L(1)\) 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 \( 1 + T \)
5 \( 1 + iT \)
good7 \( 1 - 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 2T + T^{2} \)
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

−8.887484216916158388395060563386, −8.316959596523776567222883685941, −7.33438178128938753698114340475, −6.36681335763592780053856351684, −5.75349590667892268663322316178, −5.16917982287412399886064201762, −4.70531615339593748017124364836, −3.47293448985008455850743269577, −2.25826098126639690136700412319, −1.33672138886155401013222576204, 0.48438635220208810746647840892, 1.74734552763181374110662214763, 3.13723728411488524040092425459, 4.11202120942474389035166806814, 4.43371192726024344295222772254, 5.67554635948981915123587390057, 6.37512496409794968348021218976, 7.03095272785522171683243936835, 7.45970949274410699590909001572, 8.162962266682954535427776334129

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