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Properties

Label 2-1680-5.4-c1-0-28
Degree $2$
Conductor $1680$
Sign $0.139 + 0.990i$
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 + (0.311 + 2.21i)5-s − i·7-s − 9-s − 2·11-s − 6.42i·13-s + (−2.21 + 0.311i)15-s − 4.42i·17-s − 2.42·19-s + 21-s − 1.37i·23-s + (−4.80 + 1.37i)25-s − i·27-s − 0.755·29-s − 5.18·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.139 + 0.990i)5-s − 0.377i·7-s − 0.333·9-s − 0.603·11-s − 1.78i·13-s + (−0.571 + 0.0803i)15-s − 1.07i·17-s − 0.557·19-s + 0.218·21-s − 0.287i·23-s + (−0.961 + 0.275i)25-s − 0.192i·27-s − 0.140·29-s − 0.931·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.139 + 0.990i$
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: \((2,\ 1680,\ (\ :1/2),\ 0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8394989210\)
\(L(\frac12)\) \(\approx\) \(0.8394989210\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 - 2.21i)T \)
7 \( 1 + iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6.42iT - 13T^{2} \)
17 \( 1 + 4.42iT - 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 + 1.37iT - 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 + 7.61iT - 37T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 - 2.75iT - 47T^{2} \)
53 \( 1 - 9.18iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 - 2.75iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 1.57iT - 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−9.303100583719757210198491439011, −8.339966104612306306350756484792, −7.51639021233148224979430610947, −6.91502716857745439398581482994, −5.71039441921052383012876860210, −5.24973215276669248711365590235, −3.97583967422781439248899551662, −3.14715236085892919591777443794, −2.35608963455564249349736641983, −0.30448326162175073305230041462, 1.50625122641719413256357108992, 2.18634632881611606317491414288, 3.70523218506629136680591324629, 4.65112403565697059063266718931, 5.47658532352634950038061459233, 6.35533725146129395319143987885, 7.05752380221728339180934203409, 8.283861653990198864381255800090, 8.496461260787694253699775505334, 9.433792603250466096326390515024

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