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Properties

Label 2-1911-1911.1544-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.892 + 0.451i$
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  + (−0.623 + 0.781i)3-s + (−0.0747 − 0.997i)4-s + (−0.733 − 0.680i)7-s + (−0.222 − 0.974i)9-s + (0.826 + 0.563i)12-s + (−0.365 + 0.930i)13-s + (−0.988 + 0.149i)16-s − 0.589i·19-s + (0.988 − 0.149i)21-s + (−0.955 − 0.294i)25-s + (0.900 + 0.433i)27-s + (−0.623 + 0.781i)28-s + (−0.751 − 0.433i)31-s + (−0.955 + 0.294i)36-s + (−1.98 − 0.149i)37-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (−0.0747 − 0.997i)4-s + (−0.733 − 0.680i)7-s + (−0.222 − 0.974i)9-s + (0.826 + 0.563i)12-s + (−0.365 + 0.930i)13-s + (−0.988 + 0.149i)16-s − 0.589i·19-s + (0.988 − 0.149i)21-s + (−0.955 − 0.294i)25-s + (0.900 + 0.433i)27-s + (−0.623 + 0.781i)28-s + (−0.751 − 0.433i)31-s + (−0.955 + 0.294i)36-s + (−1.98 − 0.149i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2554265701\)
\(L(\frac12)\) \(\approx\) \(0.2554265701\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
good2 \( 1 + (0.0747 + 0.997i)T^{2} \)
5 \( 1 + (0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + 0.589iT - T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.365 - 0.930i)T^{2} \)
31 \( 1 + (0.751 + 0.433i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.98 + 0.149i)T + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.955 + 0.294i)T^{2} \)
43 \( 1 + (1.88 - 0.284i)T + (0.955 - 0.294i)T^{2} \)
47 \( 1 + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.733 - 0.680i)T^{2} \)
61 \( 1 + (-1.48 + 0.716i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + 1.56iT - T^{2} \)
71 \( 1 + (0.365 - 0.930i)T^{2} \)
73 \( 1 + (0.332 - 1.07i)T + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.826 + 0.563i)T^{2} \)
97 \( 1 + (0.510 + 0.294i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.419301557853468140536983797522, −8.583395764307631699550581056508, −7.09407491720701784824014665072, −6.64882177881371717520806703606, −5.81043921593785151361333183387, −4.99727482407788098781482538296, −4.26080092659465694972175453363, −3.38354604140620559258915959656, −1.82733505630808133116910304597, −0.18991036740644815468487465592, 1.90711751711823267044453284062, 2.94243834651130453667694043965, 3.76317401535868919971532820631, 5.16650372302155104151406673798, 5.69788851605932781865532190452, 6.77112808150367588508172831581, 7.24947159861380701010807329782, 8.206650646278530786666907417682, 8.635005581087818396938725206884, 9.763686971424805739057601029906

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