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Properties

Label 2-1650-5.4-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + 4i·7-s + i·8-s − 9-s − 11-s + i·12-s − 6i·13-s + 4·14-s + 16-s − 2i·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 + 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.288i·12-s − 1.66i·13-s + 1.06·14-s + 0.250·16-s − 0.485i·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: \(1\)
Selberg data: \((2,\ 1650,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4iT - 7T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−8.981352038600481740689080715191, −8.058131967779387883020981639814, −7.54347591747245298412792612523, −6.08376500651138318940357171028, −5.64521412953945541145809675061, −4.76590854924204863110357672172, −3.27166121372302567731060941628, −2.65628972583205912832328670495, −1.65001841672879685493273212280, 0, 1.78032040572762581040858560484, 3.48955354535357901536994797041, 4.28981941884877696066351771314, 4.72741729625368473557963500877, 6.01768084346551104433706628457, 6.78989845527356060266276113225, 7.36056275940929173931600693863, 8.385557687984348992794682633991, 8.980848877051300644645337099655

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