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Properties

Label 2-1680-21.20-c1-0-10
Degree $2$
Conductor $1680$
Sign $-0.923 - 0.382i$
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  + (0.462 + 1.66i)3-s + 5-s + (−1.62 + 2.08i)7-s + (−2.57 + 1.54i)9-s − 0.196i·11-s − 2.37i·13-s + (0.462 + 1.66i)15-s + 3.36·17-s + 2.89i·19-s + (−4.23 − 1.75i)21-s + 5.80i·23-s + 25-s + (−3.76 − 3.57i)27-s + 5.73i·29-s + 4.66i·31-s + ⋯
L(s)  = 1  + (0.267 + 0.963i)3-s + 0.447·5-s + (−0.615 + 0.788i)7-s + (−0.857 + 0.514i)9-s − 0.0591i·11-s − 0.659i·13-s + (0.119 + 0.430i)15-s + 0.817·17-s + 0.663i·19-s + (−0.923 − 0.382i)21-s + 1.20i·23-s + 0.200·25-s + (−0.725 − 0.688i)27-s + 1.06i·29-s + 0.838i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320398713\)
\(L(\frac12)\) \(\approx\) \(1.320398713\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.462 - 1.66i)T \)
5 \( 1 - T \)
7 \( 1 + (1.62 - 2.08i)T \)
good11 \( 1 + 0.196iT - 11T^{2} \)
13 \( 1 + 2.37iT - 13T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 - 5.80iT - 23T^{2} \)
29 \( 1 - 5.73iT - 29T^{2} \)
31 \( 1 - 4.66iT - 31T^{2} \)
37 \( 1 + 7.34T + 37T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 - 0.444T + 43T^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 - 2.24iT - 53T^{2} \)
59 \( 1 + 4.10T + 59T^{2} \)
61 \( 1 + 1.24iT - 61T^{2} \)
67 \( 1 + 7.26T + 67T^{2} \)
71 \( 1 - 8.21iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 - 9.71T + 79T^{2} \)
83 \( 1 - 5.51T + 83T^{2} \)
89 \( 1 + 2.30T + 89T^{2} \)
97 \( 1 - 6.59iT - 97T^{2} \)
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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

−9.794862313720993876837731515826, −8.959714094974240836134371908745, −8.402014715561933853682951925170, −7.40958844130261425337056706184, −6.24990936409462599357004424671, −5.47086778942264046538477627334, −5.00111182264395525927301083859, −3.42299179291175501430761992628, −3.19767067086932745788953378903, −1.77129736627497800564823444431, 0.46299535762810615892642139217, 1.75280008234679387111366874071, 2.79807275680249662112405891222, 3.77108851093074476798414924491, 4.91261942040909386592399221041, 6.08957092708102614772605102347, 6.63212157901829098687381882637, 7.31326314675354965955300436706, 8.141070206564842753545214809076, 8.973846182241420170257751194521

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