Properties

Label 2-1650-5.4-c1-0-10
Degree $2$
Conductor $1650$
Sign $-0.447 - 0.894i$
Analytic cond. $13.1753$
Root an. cond. $3.62978$
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·2-s + i·3-s − 4-s − 6-s − 2i·7-s i·8-s − 9-s − 11-s i·12-s − 4i·13-s + 2·14-s + 16-s + 6i·17-s i·18-s + 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.301·11-s − 0.288i·12-s − 1.10i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(13.1753\)
Root analytic conductor: \(3.62978\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1650} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1650,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398454960\)
\(L(\frac12)\) \(\approx\) \(1.398454960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.815626142798945392344320931911, −8.697682676434650456013254565807, −7.898714789356682017862785155492, −7.47056351506611648043844488695, −6.25910130489876625955844609685, −5.62626800481337350779750276752, −4.74478526447514835009412810378, −3.83998232499477175671902486222, −3.02920176134488095376951961213, −1.14299162909676704652205455168, 0.62995539807950837900953515777, 2.13343775033899613311035906501, 2.68733075069823990382751008584, 3.94382568015696916455588784502, 5.00495502119669651819120876736, 5.72604065846186135363486619021, 6.81727640020239548589089864255, 7.50472410226835804207180047752, 8.539656211641887564007120877719, 9.161493250100856088232914073754

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