Properties

Label 2-47190-1.1-c1-0-22
Degree $2$
Conductor $47190$
Sign $-1$
Analytic cond. $376.814$
Root an. cond. $19.4116$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 2·14-s − 15-s + 16-s − 18-s − 2·19-s − 20-s − 2·21-s − 6·23-s − 24-s + 25-s + 26-s + 27-s − 2·28-s + 30-s − 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47190\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(376.814\)
Root analytic conductor: \(19.4116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47190} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47190,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.72004698344252, −14.52753831862184, −13.94018495149329, −13.13399792177558, −12.80201245037024, −12.32687440679529, −11.68705361871994, −11.19982465870188, −10.53584046051900, −10.09622425021871, −9.519686984076151, −9.181752885249687, −8.502683179009199, −8.047937167740602, −7.504043810984193, −7.065552907395597, −6.310869591732883, −5.968169369795323, −5.061008360270063, −4.271860952752683, −3.730352915701467, −3.138359904942572, −2.395605675438167, −1.874603499123495, −0.8151712429319077, 0, 0.8151712429319077, 1.874603499123495, 2.395605675438167, 3.138359904942572, 3.730352915701467, 4.271860952752683, 5.061008360270063, 5.968169369795323, 6.310869591732883, 7.065552907395597, 7.504043810984193, 8.047937167740602, 8.502683179009199, 9.181752885249687, 9.519686984076151, 10.09622425021871, 10.53584046051900, 11.19982465870188, 11.68705361871994, 12.32687440679529, 12.80201245037024, 13.13399792177558, 13.94018495149329, 14.52753831862184, 14.72004698344252

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