Properties

Label 2-1680-7.6-c2-0-5
Degree $2$
Conductor $1680$
Sign $-0.877 - 0.480i$
Analytic cond. $45.7766$
Root an. cond. $6.76584$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 2.23i·5-s + (−6.13 − 3.36i)7-s − 2.99·9-s + 2.58·11-s − 0.0498i·13-s + 3.87·15-s + 14.2i·17-s − 14.9i·19-s + (5.82 − 10.6i)21-s + 22.2·23-s − 5.00·25-s − 5.19i·27-s + 17.4·29-s + 6.36i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s + (−0.877 − 0.480i)7-s − 0.333·9-s + 0.235·11-s − 0.00383i·13-s + 0.258·15-s + 0.835i·17-s − 0.785i·19-s + (0.277 − 0.506i)21-s + 0.968·23-s − 0.200·25-s − 0.192i·27-s + 0.600·29-s + 0.205i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.877 - 0.480i$
Analytic conductor: \(45.7766\)
Root analytic conductor: \(6.76584\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1),\ -0.877 - 0.480i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4674795003\)
\(L(\frac12)\) \(\approx\) \(0.4674795003\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 + (6.13 + 3.36i)T \)
good11 \( 1 - 2.58T + 121T^{2} \)
13 \( 1 + 0.0498iT - 169T^{2} \)
17 \( 1 - 14.2iT - 289T^{2} \)
19 \( 1 + 14.9iT - 361T^{2} \)
23 \( 1 - 22.2T + 529T^{2} \)
29 \( 1 - 17.4T + 841T^{2} \)
31 \( 1 - 6.36iT - 961T^{2} \)
37 \( 1 + 7.14T + 1.36e3T^{2} \)
41 \( 1 + 74.5iT - 1.68e3T^{2} \)
43 \( 1 + 79.2T + 1.84e3T^{2} \)
47 \( 1 - 81.3iT - 2.20e3T^{2} \)
53 \( 1 + 67.1T + 2.80e3T^{2} \)
59 \( 1 - 4.33iT - 3.48e3T^{2} \)
61 \( 1 - 109. iT - 3.72e3T^{2} \)
67 \( 1 - 49.1T + 4.48e3T^{2} \)
71 \( 1 + 97.3T + 5.04e3T^{2} \)
73 \( 1 + 116. iT - 5.32e3T^{2} \)
79 \( 1 + 98.8T + 6.24e3T^{2} \)
83 \( 1 - 112. iT - 6.88e3T^{2} \)
89 \( 1 - 77.7iT - 7.92e3T^{2} \)
97 \( 1 - 109. iT - 9.40e3T^{2} \)
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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

−9.350089252854910334530632810299, −8.960891886610361002549758168044, −8.026678647444090348413430218571, −6.99438293642303732010015979334, −6.32879443537869681244300089812, −5.32958956026806096167428182621, −4.47774265387992716715928721529, −3.65760340380165812492779254405, −2.76524226786878956038381465367, −1.20585741374104561937971404410, 0.13104080208839764914928281090, 1.60398512784772725357305637459, 2.83040975513436376799153562127, 3.40606542006564738183278397854, 4.79105662103403233330851742699, 5.74595101201108746600257631900, 6.59361741549884158757466563097, 7.00858964224528911741185165065, 8.058403173922793500974923589212, 8.774514137016871797216449435545

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