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Properties

Label 2-1680-20.3-c1-0-27
Degree $2$
Conductor $1680$
Sign $0.532 + 0.846i$
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 + (1.85 − 1.24i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s − 1.41i·11-s + (4.28 + 4.28i)13-s + (0.431 − 2.19i)15-s + (−0.944 + 0.944i)17-s + 3.31·19-s − 1.00·21-s + (3.01 − 3.01i)23-s + (1.89 − 4.62i)25-s + (−0.707 − 0.707i)27-s + 6.22i·29-s − 1.49i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.830 − 0.557i)5-s + (−0.267 − 0.267i)7-s − 0.333i·9-s − 0.426i·11-s + (1.18 + 1.18i)13-s + (0.111 − 0.566i)15-s + (−0.228 + 0.228i)17-s + 0.760·19-s − 0.218·21-s + (0.628 − 0.628i)23-s + (0.378 − 0.925i)25-s + (−0.136 − 0.136i)27-s + 1.15i·29-s − 0.268i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.532 + 0.846i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.532 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.416330668\)
\(L(\frac12)\) \(\approx\) \(2.416330668\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 + 1.24i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-4.28 - 4.28i)T + 13iT^{2} \)
17 \( 1 + (0.944 - 0.944i)T - 17iT^{2} \)
19 \( 1 - 3.31T + 19T^{2} \)
23 \( 1 + (-3.01 + 3.01i)T - 23iT^{2} \)
29 \( 1 - 6.22iT - 29T^{2} \)
31 \( 1 + 1.49iT - 31T^{2} \)
37 \( 1 + (-6.31 + 6.31i)T - 37iT^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + (0.397 - 0.397i)T - 43iT^{2} \)
47 \( 1 + (4.48 + 4.48i)T + 47iT^{2} \)
53 \( 1 + (5.22 + 5.22i)T + 53iT^{2} \)
59 \( 1 - 0.686T + 59T^{2} \)
61 \( 1 + 2.34T + 61T^{2} \)
67 \( 1 + (7.14 + 7.14i)T + 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-4.76 - 4.76i)T + 73iT^{2} \)
79 \( 1 + 5.33T + 79T^{2} \)
83 \( 1 + (6.21 - 6.21i)T - 83iT^{2} \)
89 \( 1 + 2.76iT - 89T^{2} \)
97 \( 1 + (8.09 - 8.09i)T - 97iT^{2} \)
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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

−9.048355514848159905754214370453, −8.693694335671813451214792511964, −7.68645755083675565504215813432, −6.64471828581104100337680357235, −6.20294517047113483960119109724, −5.17415059400692864831176582537, −4.14215922366506636782859872506, −3.16495614499169194726720929623, −1.95408415085136520206416190459, −1.01171209186827277350851480347, 1.36938044365549323357548010671, 2.75706010139993778341350815451, 3.24730609997366216146939914879, 4.48746133800419308695760591072, 5.57352289802339255175557109864, 6.08838618124740397025080877889, 7.11407741656242478312532377979, 7.957370218785627265198293497796, 8.792447431301104792371316513910, 9.679750245206079909800349180235

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