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Properties

Label 2-1680-60.59-c1-0-54
Degree $2$
Conductor $1680$
Sign $-0.255 + 0.966i$
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  + (−1.64 + 0.531i)3-s + (1.20 − 1.88i)5-s − 7-s + (2.43 − 1.75i)9-s + 4.93·11-s − 6.01i·13-s + (−0.990 + 3.74i)15-s − 3.98·17-s − 1.78i·19-s + (1.64 − 0.531i)21-s + 1.63i·23-s + (−2.08 − 4.54i)25-s + (−3.08 + 4.18i)27-s + 1.89i·29-s + 4.73i·31-s + ⋯
L(s)  = 1  + (−0.951 + 0.306i)3-s + (0.540 − 0.841i)5-s − 0.377·7-s + (0.811 − 0.583i)9-s + 1.48·11-s − 1.66i·13-s + (−0.255 + 0.966i)15-s − 0.965·17-s − 0.410i·19-s + (0.359 − 0.115i)21-s + 0.341i·23-s + (−0.416 − 0.909i)25-s + (−0.593 + 0.804i)27-s + 0.352i·29-s + 0.850i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.255 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094916559\)
\(L(\frac12)\) \(\approx\) \(1.094916559\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.531i)T \)
5 \( 1 + (-1.20 + 1.88i)T \)
7 \( 1 + T \)
good11 \( 1 - 4.93T + 11T^{2} \)
13 \( 1 + 6.01iT - 13T^{2} \)
17 \( 1 + 3.98T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 - 1.63iT - 23T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 - 4.73iT - 31T^{2} \)
37 \( 1 - 6.52iT - 37T^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 - 1.57T + 43T^{2} \)
47 \( 1 + 8.90iT - 47T^{2} \)
53 \( 1 - 0.950T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 8.59T + 61T^{2} \)
67 \( 1 + 9.74T + 67T^{2} \)
71 \( 1 + 3.36T + 71T^{2} \)
73 \( 1 - 7.68iT - 73T^{2} \)
79 \( 1 + 16.2iT - 79T^{2} \)
83 \( 1 + 0.520iT - 83T^{2} \)
89 \( 1 + 16.1iT - 89T^{2} \)
97 \( 1 + 8.58iT - 97T^{2} \)
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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

−9.059022776198459793257600653974, −8.698786027357841239401331927550, −7.33094751904922135330645781894, −6.51388272127845197471646710687, −5.83367970334950095436929505445, −5.08894916826923152630833863673, −4.28112563503204592828010156084, −3.25930982530738592064069893347, −1.60289897995000403694264530350, −0.49883402267169722735428444117, 1.44529351761405440647065968247, 2.36641711636792987741802349363, 3.91197191982519655246038666438, 4.51638440690830126765698658061, 5.90353377350427462570547114082, 6.48096772664642714120823929746, 6.77326298929765190950527834207, 7.73493164469445774725671489852, 9.217994444163544130015269224368, 9.427888425956445100026480729994

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