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Properties

Label 2-1911-1911.1202-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.980 - 0.195i$
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  + (−0.974 − 0.222i)3-s + (0.930 − 0.365i)4-s + (0.826 + 0.563i)7-s + (0.900 + 0.433i)9-s + (−0.988 + 0.149i)12-s + (−0.294 + 0.955i)13-s + (0.733 − 0.680i)16-s + (0.922 + 0.922i)19-s + (−0.680 − 0.733i)21-s + (−0.997 + 0.0747i)25-s + (−0.781 − 0.623i)27-s + (0.974 + 0.222i)28-s + (−0.514 + 1.91i)31-s + (0.997 + 0.0747i)36-s + (0.733 − 0.319i)37-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)3-s + (0.930 − 0.365i)4-s + (0.826 + 0.563i)7-s + (0.900 + 0.433i)9-s + (−0.988 + 0.149i)12-s + (−0.294 + 0.955i)13-s + (0.733 − 0.680i)16-s + (0.922 + 0.922i)19-s + (−0.680 − 0.733i)21-s + (−0.997 + 0.0747i)25-s + (−0.781 − 0.623i)27-s + (0.974 + 0.222i)28-s + (−0.514 + 1.91i)31-s + (0.997 + 0.0747i)36-s + (0.733 − 0.319i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168096745\)
\(L(\frac12)\) \(\approx\) \(1.168096745\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 + 0.222i)T \)
7 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (0.294 - 0.955i)T \)
good2 \( 1 + (-0.930 + 0.365i)T^{2} \)
5 \( 1 + (0.997 - 0.0747i)T^{2} \)
11 \( 1 + (-0.781 - 0.623i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.922 - 0.922i)T + iT^{2} \)
23 \( 1 + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (-0.955 + 0.294i)T^{2} \)
31 \( 1 + (0.514 - 1.91i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.733 + 0.319i)T + (0.680 - 0.733i)T^{2} \)
41 \( 1 + (0.997 - 0.0747i)T^{2} \)
43 \( 1 + (1.35 + 1.46i)T + (-0.0747 + 0.997i)T^{2} \)
47 \( 1 + (0.149 - 0.988i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.563 + 0.826i)T^{2} \)
61 \( 1 + (-1.54 + 1.23i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-1.19 + 1.19i)T - iT^{2} \)
71 \( 1 + (-0.294 + 0.955i)T^{2} \)
73 \( 1 + (1.21 + 1.04i)T + (0.149 + 0.988i)T^{2} \)
79 \( 1 + (0.781 - 1.35i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.781 - 0.623i)T^{2} \)
89 \( 1 + (-0.149 - 0.988i)T^{2} \)
97 \( 1 + (-0.392 + 1.46i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.685360872128638640255635438683, −8.540486647410381272861859441313, −7.59477536472169604474038317616, −7.02347506832263915331154968782, −6.20358572659257558357260233707, −5.44395516474416143586671909626, −4.90429447244155168776807021919, −3.60449225855786824964262798886, −2.09378569049259112846195417872, −1.49511219042631435300128125006, 1.09144682440682462806166882766, 2.40718534837679508973013219686, 3.61812374647445336471796882643, 4.53754262373002244660213439773, 5.45441291899377216153379592228, 6.11312494480524971289162250292, 7.16065627389506939309288011881, 7.56423844261843604818793888134, 8.345579937361333059280965867772, 9.760478299315831591111495967155

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