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Properties

Label 2-1680-35.4-c1-0-44
Degree $2$
Conductor $1680$
Sign $-0.837 + 0.546i$
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  + (−0.866 − 0.5i)3-s + (1.55 − 1.61i)5-s + (−0.933 − 2.47i)7-s + (0.499 + 0.866i)9-s + (2.87 − 4.98i)11-s + 4.56i·13-s + (−2.14 + 0.620i)15-s + (−4.23 − 2.44i)17-s + (−2.68 − 4.64i)19-s + (−0.429 + 2.61i)21-s + (0.775 − 0.447i)23-s + (−0.193 − 4.99i)25-s − 0.999i·27-s + 4.20·29-s + (−1.74 + 3.01i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.693 − 0.720i)5-s + (−0.352 − 0.935i)7-s + (0.166 + 0.288i)9-s + (0.867 − 1.50i)11-s + 1.26i·13-s + (−0.554 + 0.160i)15-s + (−1.02 − 0.593i)17-s + (−0.614 − 1.06i)19-s + (−0.0936 + 0.569i)21-s + (0.161 − 0.0933i)23-s + (−0.0386 − 0.999i)25-s − 0.192i·27-s + 0.780·29-s + (−0.312 + 0.541i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.234506591\)
\(L(\frac12)\) \(\approx\) \(1.234506591\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.55 + 1.61i)T \)
7 \( 1 + (0.933 + 2.47i)T \)
good11 \( 1 + (-2.87 + 4.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
17 \( 1 + (4.23 + 2.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.68 + 4.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.775 + 0.447i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.20T + 29T^{2} \)
31 \( 1 + (1.74 - 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.89 + 2.25i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.67T + 41T^{2} \)
43 \( 1 - 8.57iT - 43T^{2} \)
47 \( 1 + (3.14 - 1.81i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.9 + 6.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.50 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.04 - 5.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.75 + 4.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.38T + 71T^{2} \)
73 \( 1 + (7.10 + 4.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.64 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.44iT - 83T^{2} \)
89 \( 1 + (1.88 + 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.32iT - 97T^{2} \)
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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

−9.116303436924735096682186230561, −8.431418697157736502832180449625, −7.20235298244416813333238372601, −6.41871059000923745485672012965, −6.11627123424963416551726655440, −4.70299105147821445958916500501, −4.30445810801561031244448188952, −2.88703685777530016685774779586, −1.51373572436653286951039776152, −0.50331475479368600623764381701, 1.74032205979392983321897585759, 2.64434931503600469403047518917, 3.82219095420158357584740634676, 4.80579234634737748968329834122, 5.85724850625034385650007758550, 6.26821476735785066068787053194, 7.07073806595901304735958380820, 8.117036361821718242057227422471, 9.138887204785666773955174008258, 9.746375102471771201810633172219

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