Properties

Label 2-1680-420.59-c0-0-3
Degree $2$
Conductor $1680$
Sign $0.605 + 0.795i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
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 + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 9-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)21-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 27-s − 1.73i·29-s − 0.999·35-s + 41-s − 43-s + (−0.5 − 0.866i)45-s + (1 + 1.73i)47-s + ⋯
L(s)  = 1  + 3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + 9-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)21-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 27-s − 1.73i·29-s − 0.999·35-s + 41-s − 43-s + (−0.5 − 0.866i)45-s + (1 + 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520962514\)
\(L(\frac12)\) \(\approx\) \(1.520962514\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.454773608195679887008379194942, −8.470009107532137536614341924555, −7.83310387459976979389831335563, −7.52274560645316690753108591571, −6.28725570900515319270898399069, −5.10049163080790931526605806148, −4.12751978795193242312977588963, −3.80711017784636327944538132443, −2.32357199325351873157807226934, −1.17896150314427398360588005636, 1.90310098412883740314056132220, 2.70631165555979516421769316171, 3.62454264742552273014860565001, 4.48920987627078146254991613648, 5.62703403777014058933266887230, 6.68031155306567719063254269574, 7.35719085514996154521453302151, 8.276129726807801814644861045428, 8.604610452302635385444119066389, 9.607458509375348379568536322599

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