Properties

Label 2-1911-273.215-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.996 + 0.0890i$
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  + i·3-s + (−0.866 − 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−1.36 + 1.36i)19-s + (−0.866 + 0.5i)25-s i·27-s + (−1.36 − 0.366i)31-s + (0.866 + 0.5i)36-s + (0.5 + 0.133i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.499i)48-s + ⋯
L(s)  = 1  + i·3-s + (−0.866 − 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−1.36 + 1.36i)19-s + (−0.866 + 0.5i)25-s i·27-s + (−1.36 − 0.366i)31-s + (0.866 + 0.5i)36-s + (0.5 + 0.133i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.499i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.996 + 0.0890i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.996 + 0.0890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2065742741\)
\(L(\frac12)\) \(\approx\) \(0.2065742741\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.762952571265420112150601943243, −9.245222113720147819105767276661, −8.337051700387611577796610593061, −7.77485456718692723425788698450, −6.32002768437934809944348336284, −5.63125842014480916359814700242, −4.91462636011125901705060056737, −4.12102083222920246905467550340, −3.38228421077878209110660240298, −1.94045938752911281675430471050, 0.14485906872427889621092487023, 1.94568860600968067248310252991, 2.89103132590686388915593419621, 4.07576047968334170661709160599, 4.89760807258971427383140061147, 5.81973955234642667384501560513, 6.88005798180805097150728296988, 7.35056009940138684526931953451, 8.295884762509138737817817846094, 8.829375789260086540890173838947

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