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Properties

Label 2-1680-105.104-c1-0-73
Degree $2$
Conductor $1680$
Sign $-0.292 + 0.956i$
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  + (0.420 − 1.68i)3-s + (1.95 + 1.08i)5-s + (−2.37 + 1.16i)7-s + (−2.64 − 1.41i)9-s − 2.82i·11-s + 0.841·13-s + (2.64 − 2.82i)15-s + 1.19i·17-s − 4.55i·19-s + (0.955 + 4.48i)21-s + 3.29·23-s + (2.64 + 4.24i)25-s + (−3.48 + 3.85i)27-s − 7.98i·29-s − 5.53i·31-s + ⋯
L(s)  = 1  + (0.242 − 0.970i)3-s + (0.874 + 0.485i)5-s + (−0.898 + 0.439i)7-s + (−0.881 − 0.471i)9-s − 0.852i·11-s + 0.233·13-s + (0.683 − 0.730i)15-s + 0.288i·17-s − 1.04i·19-s + (0.208 + 0.978i)21-s + 0.686·23-s + (0.529 + 0.848i)25-s + (−0.671 + 0.740i)27-s − 1.48i·29-s − 0.993i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.637890147\)
\(L(\frac12)\) \(\approx\) \(1.637890147\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 + 1.68i)T \)
5 \( 1 + (-1.95 - 1.08i)T \)
7 \( 1 + (2.37 - 1.16i)T \)
good11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 0.841T + 13T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 + 7.98iT - 29T^{2} \)
31 \( 1 + 5.53iT - 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 4.65iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 3.06T + 73T^{2} \)
79 \( 1 - 7.29T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 8.11T + 97T^{2} \)
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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

−9.085385967057763042421757402948, −8.435875842115632266336870661229, −7.34624845449510906813600839243, −6.66162173587244035554981923006, −5.99434154414992433727417497502, −5.46135135929331784787840499661, −3.74188102970020629178575317736, −2.79815232537328514706616540595, −2.15298113557919333740981968549, −0.60026488755306367198527315262, 1.46593143803194734622414386549, 2.83707486562257675977582901638, 3.64675126216939752515230312200, 4.73484964068421952514382863250, 5.30330051372364206108930929560, 6.33394876890234125723586704079, 7.07864299695932050250669420410, 8.286242426480745862673370655258, 9.004520671712486316433144804103, 9.619289743703043756699753899020

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