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Properties

Label 2-2169-241.240-c1-0-94
Degree $2$
Conductor $2169$
Sign $0.935 + 0.352i$
Analytic cond. $17.3195$
Root an. cond. $4.16167$
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  + 2.39·2-s + 3.75·4-s + 3.77·5-s − 3.57i·7-s + 4.22·8-s + 9.06·10-s − 3.57i·11-s + 5.11i·13-s − 8.56i·14-s + 2.61·16-s + 2.92i·17-s + 5.99i·19-s + 14.1·20-s − 8.57i·22-s + 4.03i·23-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.87·4-s + 1.68·5-s − 1.34i·7-s + 1.49·8-s + 2.86·10-s − 1.07i·11-s + 1.41i·13-s − 2.28i·14-s + 0.653·16-s + 0.708i·17-s + 1.37i·19-s + 3.17·20-s − 1.82i·22-s + 0.841i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(17.3195\)
Root analytic conductor: \(4.16167\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2169} (1927, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2169,\ (\ :1/2),\ 0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(6.420076295\)
\(L(\frac12)\) \(\approx\) \(6.420076295\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 + (-14.5 - 5.47i)T \)
good2 \( 1 - 2.39T + 2T^{2} \)
5 \( 1 - 3.77T + 5T^{2} \)
7 \( 1 + 3.57iT - 7T^{2} \)
11 \( 1 + 3.57iT - 11T^{2} \)
13 \( 1 - 5.11iT - 13T^{2} \)
17 \( 1 - 2.92iT - 17T^{2} \)
19 \( 1 - 5.99iT - 19T^{2} \)
23 \( 1 - 4.03iT - 23T^{2} \)
29 \( 1 + 5.15T + 29T^{2} \)
31 \( 1 + 8.57iT - 31T^{2} \)
37 \( 1 + 4.52iT - 37T^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 + 5.60iT - 43T^{2} \)
47 \( 1 + 9.50T + 47T^{2} \)
53 \( 1 + 9.20T + 53T^{2} \)
59 \( 1 - 2.87T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 7.24T + 67T^{2} \)
71 \( 1 + 9.84iT - 71T^{2} \)
73 \( 1 - 3.53iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−9.311463598946648670008348701523, −8.011429585319869430648251738058, −7.03407798109689146441964238242, −6.26401090844792178567377458736, −5.88831788344044116476191218794, −5.11258594750217065547066967679, −3.95098971008640173624377342044, −3.65315676670603067127727290407, −2.22527731479475467610669211764, −1.51303136090768976310009336687, 1.76377656652059979951616698949, 2.67315723030003089355348838122, 2.96364815988221281418797007310, 4.69689144038659088025058669190, 5.17113782850181471951970176497, 5.64736385142895666254313685880, 6.47733865986968503086809071595, 7.01807529409922203886032449836, 8.363847859907715281728179692647, 9.323983186869228014836483884957

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