Properties

Label 2-1680-5.4-c1-0-34
Degree $2$
Conductor $1680$
Sign $-0.894 + 0.447i$
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 + (2 − i)5-s i·7-s − 9-s − 4·11-s − 2i·13-s + (−1 − 2i)15-s + 2i·17-s − 2·19-s − 21-s − 6i·23-s + (3 − 4i)25-s + i·27-s − 6·29-s − 6·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.894 − 0.447i)5-s − 0.377i·7-s − 0.333·9-s − 1.20·11-s − 0.554i·13-s + (−0.258 − 0.516i)15-s + 0.485i·17-s − 0.458·19-s − 0.218·21-s − 1.25i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 1680,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.184505347\)
\(L(\frac12)\) \(\approx\) \(1.184505347\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
7 \( 1 + iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.892317637351689119864639239854, −8.238221615209891668067861805353, −7.46076405834372518269636748419, −6.59041345150838864876647244373, −5.68340562035333934168063884294, −5.16279447382091566237105160711, −3.96093171659562900994942581441, −2.64614280749033305500312946816, −1.84559586549780823792609125113, −0.41002701862452815638068003388, 1.84652139555756762753248856884, 2.74431134969039036229484213345, 3.70456521129806073514636610767, 5.00808821292139965791964853524, 5.48754715163448808142478910701, 6.35471301963850703238367892455, 7.31389556620405462878996554263, 8.129618539359861725128170546132, 9.309383403710638666001786121605, 9.473085835099832135961826760589

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