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Properties

Label 2-3344-836.303-c0-0-1
Degree $2$
Conductor $3344$
Sign $0.854 - 0.518i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
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  + (0.0646 − 0.198i)5-s + (−1.47 + 1.07i)7-s + (−0.309 − 0.951i)9-s + (0.913 − 0.406i)11-s + (−0.395 − 0.128i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (0.773 + 0.562i)25-s + (0.118 + 0.363i)35-s + 1.33·43-s − 0.209·45-s + (0.873 − 1.20i)47-s + (0.722 − 2.22i)49-s + (−0.0218 − 0.207i)55-s + (1.41 + 0.459i)61-s + ⋯
L(s)  = 1  + (0.0646 − 0.198i)5-s + (−1.47 + 1.07i)7-s + (−0.309 − 0.951i)9-s + (0.913 − 0.406i)11-s + (−0.395 − 0.128i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (0.773 + 0.562i)25-s + (0.118 + 0.363i)35-s + 1.33·43-s − 0.209·45-s + (0.873 − 1.20i)47-s + (0.722 − 2.22i)49-s + (−0.0218 − 0.207i)55-s + (1.41 + 0.459i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.854 - 0.518i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.854 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068826790\)
\(L(\frac12)\) \(\approx\) \(1.068826790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.0646 + 0.198i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.395 + 0.128i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 1.33T + T^{2} \)
47 \( 1 + (-0.873 + 1.20i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.41 - 0.459i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.01 - 1.40i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.114452864372361484256302155303, −8.369802977763128264870187058757, −7.11874287657605984775984650963, −6.64886050616092460887546335964, −5.65530236850934820729158466558, −5.55673700859216724033833948464, −3.85016327207691179039464622272, −3.44439131428084719565225490209, −2.54439780152644602588247411894, −1.09443185962745735857608954398, 0.77931997983078347239905114304, 2.37619564416317180738057880263, 3.11436681694393168874660290672, 4.15726221234186593082390661934, 4.68934979789520280101016337730, 5.93446250535338114168789774046, 6.65225907346897413686755695363, 7.05932302590702917577932457600, 7.891380278991686359955112538886, 8.894136636609169758215497341694

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