Properties

Label 2-1680-5.4-c1-0-17
Degree $2$
Conductor $1680$
Sign $1$
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 + 2.23·5-s i·7-s − 9-s + 2·11-s + 4.47i·13-s − 2.23i·15-s + 6.47i·17-s + 2·19-s − 21-s + 4i·23-s + 5.00·25-s + i·27-s + 8.47·29-s + 0.472·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.999·5-s − 0.377i·7-s − 0.333·9-s + 0.603·11-s + 1.24i·13-s − 0.577i·15-s + 1.56i·17-s + 0.458·19-s − 0.218·21-s + 0.834i·23-s + 1.00·25-s + 0.192i·27-s + 1.57·29-s + 0.0847·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.166450287\)
\(L(\frac12)\) \(\approx\) \(2.166450287\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
7 \( 1 + iT \)
good11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 - 6.47iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 - 0.472T + 31T^{2} \)
37 \( 1 - 2.47iT - 37T^{2} \)
41 \( 1 + 3.52T + 41T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 + 6.47iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 1.52iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 7.52iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 3.52iT - 97T^{2} \)
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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

−9.264208481133307120037657457469, −8.667292505982898318200228799091, −7.73756605431910636080781919973, −6.62754550826837263948662540680, −6.44648988037086414665311803493, −5.40008369206042728836092980195, −4.36479531582599764469203548637, −3.32299895968630026163813232651, −1.98393875322555708005438014281, −1.31107122907419097395558461800, 0.950268170574869678856090109351, 2.55117428671047054305860631940, 3.14043716612225681638802522076, 4.59066596833036635818773195792, 5.21822241519116633952655098494, 6.02589171913680573336794383658, 6.79758228274833664663064268498, 7.87381300535201958282009287904, 8.813741533408734104095286023879, 9.365662707639186489561039894012

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