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Properties

Label 2-3840-15.14-c0-0-8
Degree 22
Conductor 38403840
Sign ii
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

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

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: ii
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3840(3329,⋅)\chi_{3840} (3329, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :0), i)(2,\ 3840,\ (\ :0),\ i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.70531445220.7053144522
L(12)L(\frac12) ≈\approx 0.70531445220.7053144522
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
5 1−iT 1 - iT
good7 1+2iT−T2 1 + 2iT - T^{2}
11 1−T2 1 - T^{2}
13 1−T2 1 - T^{2}
17 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1+2iT−T2 1 + 2iT - T^{2}
31 1+T2 1 + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+T2 1 + T^{2}
83 1+2T+T2 1 + 2T + T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line