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Properties

Label 2-1680-5.4-c1-0-19
Degree $2$
Conductor $1680$
Sign $0.970 + 0.241i$
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  − i·3-s + (2.17 + 0.539i)5-s + i·7-s − 9-s − 2·11-s + 0.921i·13-s + (0.539 − 2.17i)15-s − 1.07i·17-s + 3.07·19-s + 21-s − 2.34i·23-s + (4.41 + 2.34i)25-s + i·27-s + 6.68·29-s + 7.75·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.970 + 0.241i)5-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s + 0.255i·13-s + (0.139 − 0.560i)15-s − 0.261i·17-s + 0.706·19-s + 0.218·21-s − 0.487i·23-s + (0.883 + 0.468i)25-s + 0.192i·27-s + 1.24·29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.056626811\)
\(L(\frac12)\) \(\approx\) \(2.056626811\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.539i)T \)
7 \( 1 - iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 0.921iT - 13T^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 + 2.34iT - 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 - 6.52iT - 43T^{2} \)
47 \( 1 - 4.68iT - 47T^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 4.68iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 + 8.34T + 89T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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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

−9.337996673936694624759399173239, −8.543542163294499899502896447126, −7.69642563215899881565945137413, −6.85304416634752927023672162793, −6.10160257330762550865604344611, −5.43392783452624720888604225413, −4.47314146685261746469627808608, −2.90088565925480393859685847045, −2.37751122945413633789497289089, −1.05444459159725150700998759633, 1.02981581222319332012335970657, 2.44626544083457065456892001260, 3.35450482545903181755857206240, 4.59239213455625234961375095261, 5.20286636159279683599135014054, 6.05380486159785431226940084764, 6.87228972554830790852858784849, 7.993208909930311157811685036614, 8.614558720803389750685079642372, 9.589556146707979771429218042011

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