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Properties

Label 2-1911-273.167-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.226 + 0.973i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 − 0.866i)9-s − 0.999·12-s + (0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (0.133 + 0.5i)19-s − i·25-s − 0.999i·27-s + (1 − i)31-s + (−0.866 + 0.499i)36-s + (−1.86 − 0.5i)37-s + (0.499 − 0.866i)39-s + (−1.5 − 0.866i)43-s + (0.866 + 0.499i)48-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 − 0.866i)9-s − 0.999·12-s + (0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (0.133 + 0.5i)19-s − i·25-s − 0.999i·27-s + (1 − i)31-s + (−0.866 + 0.499i)36-s + (−1.86 − 0.5i)37-s + (0.499 − 0.866i)39-s + (−1.5 − 0.866i)43-s + (0.866 + 0.499i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.226 + 0.973i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.226 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293203086\)
\(L(\frac12)\) \(\approx\) \(1.293203086\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
13 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + iT^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.026109301795774292269736960918, −8.468228188264747940893868512173, −7.974691215508871849945019391680, −6.86360573841198743737494156142, −6.08306952369741506861937128168, −5.21941730241019079832955234894, −4.08740505058268307296504633057, −3.46414561042832516680660402608, −2.17230420663953170300620100950, −0.981357450535922551213928152899, 1.64743989991798547575778101604, 3.15445190369115272751305517357, 3.57371885838641262841545941520, 4.65397914239475892740675069145, 5.15187107237021171838363371878, 6.54599138833075433764295839345, 7.41184353981072332583148958977, 8.337058638990309702928935678317, 8.699134775729992087087782292093, 9.413571242102426480928411326891

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