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Properties

Label 2-1680-20.3-c1-0-5
Degree $2$
Conductor $1680$
Sign $-0.972 + 0.233i$
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 + (0.586 + 2.15i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s + 1.41i·11-s + (−2.65 − 2.65i)13-s + (−1.94 − 1.11i)15-s + (−0.119 + 0.119i)17-s − 4.98·19-s − 1.00·21-s + (−5.67 + 5.67i)23-s + (−4.31 + 2.52i)25-s + (0.707 + 0.707i)27-s − 1.53i·29-s + 2.65i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.262 + 0.965i)5-s + (0.267 + 0.267i)7-s − 0.333i·9-s + 0.426i·11-s + (−0.735 − 0.735i)13-s + (−0.500 − 0.286i)15-s + (−0.0290 + 0.0290i)17-s − 1.14·19-s − 0.218·21-s + (−1.18 + 1.18i)23-s + (−0.862 + 0.505i)25-s + (0.136 + 0.136i)27-s − 0.284i·29-s + 0.477i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.972 + 0.233i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.972 + 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5332486153\)
\(L(\frac12)\) \(\approx\) \(0.5332486153\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.586 - 2.15i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (2.65 + 2.65i)T + 13iT^{2} \)
17 \( 1 + (0.119 - 0.119i)T - 17iT^{2} \)
19 \( 1 + 4.98T + 19T^{2} \)
23 \( 1 + (5.67 - 5.67i)T - 23iT^{2} \)
29 \( 1 + 1.53iT - 29T^{2} \)
31 \( 1 - 2.65iT - 31T^{2} \)
37 \( 1 + (-3.16 + 3.16i)T - 37iT^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + (2.21 - 2.21i)T - 43iT^{2} \)
47 \( 1 + (-0.161 - 0.161i)T + 47iT^{2} \)
53 \( 1 + (5.20 + 5.20i)T + 53iT^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + (9.82 + 9.82i)T + 67iT^{2} \)
71 \( 1 + 6.54iT - 71T^{2} \)
73 \( 1 + (-4.45 - 4.45i)T + 73iT^{2} \)
79 \( 1 - 1.29T + 79T^{2} \)
83 \( 1 + (1.88 - 1.88i)T - 83iT^{2} \)
89 \( 1 - 4.74iT - 89T^{2} \)
97 \( 1 + (9.82 - 9.82i)T - 97iT^{2} \)
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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

−9.759040370275949375590227193763, −9.306478326630776432032112870955, −7.970941980142533475808914497777, −7.49577001991600727047392438681, −6.37121287326271377842295375913, −5.85470817437219055600548744315, −4.87575527109830272441261259301, −3.95076224630236444305168350341, −2.86977287720852119288387772008, −1.89792797811613354954448083348, 0.20216177308468406871116797418, 1.55272080534882733681455626172, 2.52823054175718555478528327344, 4.30868556926852204803039750376, 4.57635554496534512702381099067, 5.83287293884464950605663726097, 6.32088846079383029299966619593, 7.40472679803517884974084404649, 8.157447267031943729113708648008, 8.858633110572442913509034553670

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