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Properties

Label 2-2217-2217.1280-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.977 + 0.211i$
Analytic cond. $1.10642$
Root an. cond. $1.05186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.114 − 0.993i)3-s + (0.0383 + 0.999i)4-s + (−1.33 − 1.28i)7-s + (−0.973 + 0.227i)9-s + (0.988 − 0.152i)12-s + (0.456 + 0.839i)13-s + (−0.997 + 0.0765i)16-s + (−0.293 − 1.51i)19-s + (−1.12 + 1.47i)21-s + (−0.973 − 0.227i)25-s + (0.338 + 0.941i)27-s + (1.23 − 1.38i)28-s + (−0.930 − 1.71i)31-s + (−0.264 − 0.964i)36-s + (−1.63 + 0.125i)37-s + ⋯
L(s)  = 1  + (−0.114 − 0.993i)3-s + (0.0383 + 0.999i)4-s + (−1.33 − 1.28i)7-s + (−0.973 + 0.227i)9-s + (0.988 − 0.152i)12-s + (0.456 + 0.839i)13-s + (−0.997 + 0.0765i)16-s + (−0.293 − 1.51i)19-s + (−1.12 + 1.47i)21-s + (−0.973 − 0.227i)25-s + (0.338 + 0.941i)27-s + (1.23 − 1.38i)28-s + (−0.930 − 1.71i)31-s + (−0.264 − 0.964i)36-s + (−1.63 + 0.125i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3573769202\)
\(L(\frac12)\) \(\approx\) \(0.3573769202\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.114 + 0.993i)T \)
739 \( 1 + (-0.988 + 0.152i)T \)
good2 \( 1 + (-0.0383 - 0.999i)T^{2} \)
5 \( 1 + (0.973 + 0.227i)T^{2} \)
7 \( 1 + (1.33 + 1.28i)T + (0.0383 + 0.999i)T^{2} \)
11 \( 1 + (0.409 - 0.912i)T^{2} \)
13 \( 1 + (-0.456 - 0.839i)T + (-0.543 + 0.839i)T^{2} \)
17 \( 1 + (-0.720 - 0.693i)T^{2} \)
19 \( 1 + (0.293 + 1.51i)T + (-0.927 + 0.373i)T^{2} \)
23 \( 1 + (-0.0383 - 0.999i)T^{2} \)
29 \( 1 + (0.859 - 0.511i)T^{2} \)
31 \( 1 + (0.930 + 1.71i)T + (-0.543 + 0.839i)T^{2} \)
37 \( 1 + (1.63 - 0.125i)T + (0.988 - 0.152i)T^{2} \)
41 \( 1 + (-0.477 + 0.878i)T^{2} \)
43 \( 1 + (0.521 - 1.45i)T + (-0.771 - 0.636i)T^{2} \)
47 \( 1 + (0.665 - 0.746i)T^{2} \)
53 \( 1 + (-0.988 + 0.152i)T^{2} \)
59 \( 1 + (0.771 - 0.636i)T^{2} \)
61 \( 1 + (1.20 + 1.58i)T + (-0.264 + 0.964i)T^{2} \)
67 \( 1 + (0.0745 - 0.0174i)T + (0.896 - 0.443i)T^{2} \)
71 \( 1 + (0.665 + 0.746i)T^{2} \)
73 \( 1 + (1.74 + 0.864i)T + (0.606 + 0.795i)T^{2} \)
79 \( 1 + (-1.17 + 0.829i)T + (0.338 - 0.941i)T^{2} \)
83 \( 1 + (-0.338 + 0.941i)T^{2} \)
89 \( 1 + (0.543 + 0.839i)T^{2} \)
97 \( 1 + (-1.42 - 1.37i)T + (0.0383 + 0.999i)T^{2} \)
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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

−8.911346397451069724255208852132, −7.75812812201525011632090295656, −7.39497026385536632784654928546, −6.55514949313266138733284954913, −6.26592140365997515575257537590, −4.68468860685428507440339622449, −3.77044667154999495146039772721, −3.08002334975561334722516366310, −1.95452939561231655831722338133, −0.23047259107190489601323116572, 1.90489219624470809762907136255, 3.13971923709559094508914852671, 3.74878673635001665440440297579, 5.11104130241705134541578198311, 5.76548046859423887905992177347, 5.99640636573364262765290645014, 7.05771224348656421157139384209, 8.541392840369399738970920899756, 8.890193318843139728788869908607, 9.691670672098064503178368438775

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