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Properties

Label 2-1680-35.13-c1-0-4
Degree $2$
Conductor $1680$
Sign $-0.566 - 0.824i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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  + (−0.707 + 0.707i)3-s + (−1.19 − 1.88i)5-s + (2.59 + 0.510i)7-s − 1.00i·9-s − 4.79·11-s + (−0.585 + 0.585i)13-s + (2.18 + 0.489i)15-s + (4.10 + 4.10i)17-s − 2.36·19-s + (−2.19 + 1.47i)21-s + (−2.97 − 2.97i)23-s + (−2.13 + 4.52i)25-s + (0.707 + 0.707i)27-s − 9.94i·29-s + 3.02i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.535 − 0.844i)5-s + (0.981 + 0.192i)7-s − 0.333i·9-s − 1.44·11-s + (−0.162 + 0.162i)13-s + (0.563 + 0.126i)15-s + (0.995 + 0.995i)17-s − 0.542·19-s + (−0.479 + 0.321i)21-s + (−0.621 − 0.621i)23-s + (−0.427 + 0.904i)25-s + (0.136 + 0.136i)27-s − 1.84i·29-s + 0.542i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.566 - 0.824i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.566 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6045772316\)
\(L(\frac12)\) \(\approx\) \(0.6045772316\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.19 + 1.88i)T \)
7 \( 1 + (-2.59 - 0.510i)T \)
good11 \( 1 + 4.79T + 11T^{2} \)
13 \( 1 + (0.585 - 0.585i)T - 13iT^{2} \)
17 \( 1 + (-4.10 - 4.10i)T + 17iT^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 + (2.97 + 2.97i)T + 23iT^{2} \)
29 \( 1 + 9.94iT - 29T^{2} \)
31 \( 1 - 3.02iT - 31T^{2} \)
37 \( 1 + (6.10 - 6.10i)T - 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (-5.74 - 5.74i)T + 43iT^{2} \)
47 \( 1 + (-0.363 - 0.363i)T + 47iT^{2} \)
53 \( 1 + (-2.36 - 2.36i)T + 53iT^{2} \)
59 \( 1 + 2.07T + 59T^{2} \)
61 \( 1 - 5.55iT - 61T^{2} \)
67 \( 1 + (-0.979 + 0.979i)T - 67iT^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + (6.11 - 6.11i)T - 73iT^{2} \)
79 \( 1 - 5.10iT - 79T^{2} \)
83 \( 1 + (3.22 - 3.22i)T - 83iT^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (8.05 + 8.05i)T + 97iT^{2} \)
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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

−9.789949851926847839564162223802, −8.550273950978958668011192028765, −8.157097763607451937586650963800, −7.56405259985566705267896464424, −6.13916517750472802522270527349, −5.43208641061236381629251105442, −4.66121029352083023790031145487, −4.07993214234008173395743172171, −2.66935343945688264052363546363, −1.32673205022033832266734675956, 0.24956037898509668044881231487, 1.92590521224968597106288514904, 2.92925138558121036575306605209, 4.01528857812071394433852880176, 5.23696559052976933306929994677, 5.57809167006699170695950100209, 7.00562501779377451367148800273, 7.45983705476908691763044023051, 7.971130983126011037009242405478, 8.933623876867325448614321388235

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