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Properties

Label 2-1680-20.7-c1-0-8
Degree $2$
Conductor $1680$
Sign $-0.312 - 0.949i$
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 + (2.14 − 0.635i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s + 1.41i·11-s + (−4.66 + 4.66i)13-s + (1.96 + 1.06i)15-s + (3.52 + 3.52i)17-s − 7.26·19-s − 1.00·21-s + (−2.99 − 2.99i)23-s + (4.19 − 2.72i)25-s + (−0.707 + 0.707i)27-s + 7.18i·29-s + 7.80i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.958 − 0.284i)5-s + (−0.267 + 0.267i)7-s + 0.333i·9-s + 0.426i·11-s + (−1.29 + 1.29i)13-s + (0.507 + 0.275i)15-s + (0.854 + 0.854i)17-s − 1.66·19-s − 0.218·21-s + (−0.625 − 0.625i)23-s + (0.838 − 0.544i)25-s + (−0.136 + 0.136i)27-s + 1.33i·29-s + 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.312 - 0.949i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.312 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.721532241\)
\(L(\frac12)\) \(\approx\) \(1.721532241\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 0.635i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (4.66 - 4.66i)T - 13iT^{2} \)
17 \( 1 + (-3.52 - 3.52i)T + 17iT^{2} \)
19 \( 1 + 7.26T + 19T^{2} \)
23 \( 1 + (2.99 + 2.99i)T + 23iT^{2} \)
29 \( 1 - 7.18iT - 29T^{2} \)
31 \( 1 - 7.80iT - 31T^{2} \)
37 \( 1 + (-0.824 - 0.824i)T + 37iT^{2} \)
41 \( 1 + 2.09T + 41T^{2} \)
43 \( 1 + (3.56 + 3.56i)T + 43iT^{2} \)
47 \( 1 + (1.31 - 1.31i)T - 47iT^{2} \)
53 \( 1 + (-6.73 + 6.73i)T - 53iT^{2} \)
59 \( 1 - 7.63T + 59T^{2} \)
61 \( 1 - 8.13T + 61T^{2} \)
67 \( 1 + (-4.41 + 4.41i)T - 67iT^{2} \)
71 \( 1 - 9.78iT - 71T^{2} \)
73 \( 1 + (-0.731 + 0.731i)T - 73iT^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + (-0.102 - 0.102i)T + 83iT^{2} \)
89 \( 1 + 8.17iT - 89T^{2} \)
97 \( 1 + (-9.35 - 9.35i)T + 97iT^{2} \)
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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

−9.697889627679349153915380203925, −8.766407344313998964566471081082, −8.380300947376269577255374899585, −6.98009093617537702046232328886, −6.51402344966327376406665244832, −5.36196741930484058849323946599, −4.70195767920419168719829139031, −3.73924891875202576338828590580, −2.41200974780314702740482100292, −1.79347287838048810511090075809, 0.57099493712826120866042388789, 2.19250459165271927730799558040, 2.78569269896971949145891768624, 3.92074883331050500021832771545, 5.20683354241112886513151595505, 5.91515286217059129463617359665, 6.69537232414618881273283328091, 7.64071965324020766757140442013, 8.134246981397508133062598373721, 9.278329721408880850736360265522

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