Properties

Label 2-1680-5.4-c1-0-9
Degree $2$
Conductor $1680$
Sign $0.139 - 0.990i$
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 + (−0.311 + 2.21i)5-s + i·7-s − 9-s + 3.80·11-s + 0.622i·13-s + (2.21 + 0.311i)15-s + 4.42i·17-s + 0.622·19-s + 21-s − 2.62i·23-s + (−4.80 − 1.37i)25-s + i·27-s − 9.61·29-s + 0.622·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.139 + 0.990i)5-s + 0.377i·7-s − 0.333·9-s + 1.14·11-s + 0.172i·13-s + (0.571 + 0.0803i)15-s + 1.07i·17-s + 0.142·19-s + 0.218·21-s − 0.546i·23-s + (−0.961 − 0.275i)25-s + 0.192i·27-s − 1.78·29-s + 0.111·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.139 - 0.990i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.410726225\)
\(L(\frac12)\) \(\approx\) \(1.410726225\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.311 - 2.21i)T \)
7 \( 1 - iT \)
good11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 - 0.622iT - 13T^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 - 0.622T + 19T^{2} \)
23 \( 1 + 2.62iT - 23T^{2} \)
29 \( 1 + 9.61T + 29T^{2} \)
31 \( 1 - 0.622T + 31T^{2} \)
37 \( 1 - 1.24iT - 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 - 4.85iT - 43T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 8.10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + 6.75T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 + 4.23iT - 97T^{2} \)
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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

−9.425891230274500352550375464608, −8.764501356587066383827042285455, −7.77076134800962619914153566080, −7.18151277010689044655174516280, −6.19493189581919366639403922383, −5.94876919956031967515745887864, −4.38688981097203963080989528665, −3.53795236370793453551078348072, −2.50126429883998132797403839887, −1.44948508828435448526484678577, 0.55617799536496794240769176047, 1.89321449668846189195373127875, 3.50381520205702350857196039088, 4.07192536679267238635122808977, 5.06045193145988207409803390054, 5.66602528077929526853341835658, 6.85462522200396897907130113956, 7.62411692660834771829352859231, 8.570060241034099921841991493292, 9.306892179309318777849833925684

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