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Properties

Label 2-1680-20.7-c1-0-28
Degree $2$
Conductor $1680$
Sign $-0.720 + 0.693i$
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 + (1.50 + 1.65i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s − 5.47i·11-s + (−1.81 + 1.81i)13-s + (0.108 − 2.23i)15-s + (−2.81 − 2.81i)17-s − 7.29·19-s + 1.00·21-s + (3.15 + 3.15i)23-s + (−0.484 + 4.97i)25-s + (0.707 − 0.707i)27-s − 1.13i·29-s − 10.8i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.671 + 0.740i)5-s + (−0.267 + 0.267i)7-s + 0.333i·9-s − 1.65i·11-s + (−0.502 + 0.502i)13-s + (0.0280 − 0.576i)15-s + (−0.681 − 0.681i)17-s − 1.67·19-s + 0.218·21-s + (0.657 + 0.657i)23-s + (−0.0969 + 0.995i)25-s + (0.136 − 0.136i)27-s − 0.210i·29-s − 1.95i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6258562238\)
\(L(\frac12)\) \(\approx\) \(0.6258562238\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 1.65i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 5.47iT - 11T^{2} \)
13 \( 1 + (1.81 - 1.81i)T - 13iT^{2} \)
17 \( 1 + (2.81 + 2.81i)T + 17iT^{2} \)
19 \( 1 + 7.29T + 19T^{2} \)
23 \( 1 + (-3.15 - 3.15i)T + 23iT^{2} \)
29 \( 1 + 1.13iT - 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + (1.90 + 1.90i)T + 37iT^{2} \)
41 \( 1 + 7.20T + 41T^{2} \)
43 \( 1 + (0.875 + 0.875i)T + 43iT^{2} \)
47 \( 1 + (-1.75 + 1.75i)T - 47iT^{2} \)
53 \( 1 + (-4.85 + 4.85i)T - 53iT^{2} \)
59 \( 1 + 2.59T + 59T^{2} \)
61 \( 1 - 2.17T + 61T^{2} \)
67 \( 1 + (-4.36 + 4.36i)T - 67iT^{2} \)
71 \( 1 - 5.18iT - 71T^{2} \)
73 \( 1 + (-10.3 + 10.3i)T - 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (8.81 + 8.81i)T + 83iT^{2} \)
89 \( 1 + 0.630iT - 89T^{2} \)
97 \( 1 + (-0.688 - 0.688i)T + 97iT^{2} \)
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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

−9.058314561277347665996583492470, −8.331762512334076122171928839387, −7.23276914568635885152096882394, −6.53400348643151032955599646199, −5.98174249025311196762727118228, −5.19059066654171403710531004399, −3.91096854182743683776907678302, −2.78283948552484999667596453660, −1.98677422240856648512107369044, −0.23564951255645029612855227133, 1.52233731585517528557565595370, 2.57399580843736074981148010164, 4.11148915323260178973739577709, 4.71958133290076671839340746970, 5.39330923446811125366319144174, 6.58685634781808554388928233686, 6.93890287070733571661330862070, 8.327457317569057990354044096533, 8.881427066309081777017777553327, 9.799568151518343402796628340879

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