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Properties

Label 2-3311-3311.1707-c0-0-0
Degree $2$
Conductor $3311$
Sign $-0.166 + 0.985i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (−1.44 − 0.864i)2-s + (0.872 + 1.62i)4-s + (0.913 − 0.406i)7-s + (0.0635 − 1.41i)8-s + (0.925 + 0.379i)9-s + (0.550 − 0.834i)11-s + (−1.67 − 0.201i)14-s + (−0.301 + 0.456i)16-s + (−1.01 − 1.34i)18-s + (−1.51 + 0.731i)22-s + (−1.17 − 0.802i)23-s + (0.946 − 0.323i)25-s + (1.45 + 1.12i)28-s + (−0.375 − 1.90i)29-s + (−0.445 + 0.214i)32-s + ⋯
L(s)  = 1  + (−1.44 − 0.864i)2-s + (0.872 + 1.62i)4-s + (0.913 − 0.406i)7-s + (0.0635 − 1.41i)8-s + (0.925 + 0.379i)9-s + (0.550 − 0.834i)11-s + (−1.67 − 0.201i)14-s + (−0.301 + 0.456i)16-s + (−1.01 − 1.34i)18-s + (−1.51 + 0.731i)22-s + (−1.17 − 0.802i)23-s + (0.946 − 0.323i)25-s + (1.45 + 1.12i)28-s + (−0.375 − 1.90i)29-s + (−0.445 + 0.214i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $-0.166 + 0.985i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ -0.166 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7343216975\)
\(L(\frac12)\) \(\approx\) \(0.7343216975\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (-0.550 + 0.834i)T \)
43 \( 1 + (0.791 + 0.611i)T \)
good2 \( 1 + (1.44 + 0.864i)T + (0.473 + 0.880i)T^{2} \)
3 \( 1 + (-0.925 - 0.379i)T^{2} \)
5 \( 1 + (-0.946 + 0.323i)T^{2} \)
13 \( 1 + (0.992 - 0.119i)T^{2} \)
17 \( 1 + (-0.599 + 0.800i)T^{2} \)
19 \( 1 + (0.163 - 0.986i)T^{2} \)
23 \( 1 + (1.17 + 0.802i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (0.375 + 1.90i)T + (-0.925 + 0.379i)T^{2} \)
31 \( 1 + (0.999 + 0.0299i)T^{2} \)
37 \( 1 + (0.529 + 0.0556i)T + (0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.691 + 0.722i)T^{2} \)
47 \( 1 + (-0.936 + 0.351i)T^{2} \)
53 \( 1 + (-0.851 - 0.699i)T + (0.193 + 0.981i)T^{2} \)
59 \( 1 + (0.995 - 0.0896i)T^{2} \)
61 \( 1 + (0.999 - 0.0299i)T^{2} \)
67 \( 1 + (-0.0860 - 1.14i)T + (-0.988 + 0.149i)T^{2} \)
71 \( 1 + (0.254 - 0.328i)T + (-0.251 - 0.967i)T^{2} \)
73 \( 1 + (0.873 + 0.486i)T^{2} \)
79 \( 1 + (0.909 - 0.818i)T + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.525 + 0.850i)T^{2} \)
89 \( 1 + (0.0747 - 0.997i)T^{2} \)
97 \( 1 + (0.963 - 0.266i)T^{2} \)
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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

−8.511086948662842575805746792972, −8.211954361481838006583857983250, −7.43489356449554873526363616567, −6.73821348208948513532320699816, −5.63225421246775752107562721930, −4.44381945031995015947314915697, −3.81261801820896620645441575272, −2.54348710903388513976680154594, −1.75186438405932532018731754768, −0.808595954716565554915711426547, 1.37620354376791955688217452715, 1.84478331098060500979207345715, 3.55404155983771118609988602862, 4.64654386592351093217137505387, 5.41107017292026030088638088909, 6.37907405023635059353062537506, 7.07738454706205301742314825265, 7.48900522667197171872451240715, 8.319513685936849504304613591110, 8.966597331650700231240277969034

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