Properties

Label 2-47190-1.1-c1-0-20
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 − 3·7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 3·14-s − 15-s + 16-s + 3·17-s − 18-s + 4·19-s + 20-s + 3·21-s − 8·23-s + 24-s + 25-s + 26-s − 27-s − 3·28-s − 9·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.654·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s − 1.67·29-s + 0.182·30-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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.77794275997982, −14.45509322878760, −13.75065902039293, −13.23710534059764, −12.65389680483651, −12.22100752457656, −11.82045823625675, −11.08599018420705, −10.63787543525336, −9.978828007194990, −9.755423832230238, −9.216521651389946, −8.778821066020855, −7.725465366695182, −7.453088998660455, −7.020409612177150, −6.007225629374257, −5.894540931995414, −5.487336939548351, −4.390893297645096, −3.780431350529193, −3.112894224113698, −2.369318663611878, −1.668002188385914, −0.7815153769225319, 0, 0.7815153769225319, 1.668002188385914, 2.369318663611878, 3.112894224113698, 3.780431350529193, 4.390893297645096, 5.487336939548351, 5.894540931995414, 6.007225629374257, 7.020409612177150, 7.453088998660455, 7.725465366695182, 8.778821066020855, 9.216521651389946, 9.755423832230238, 9.978828007194990, 10.63787543525336, 11.08599018420705, 11.82045823625675, 12.22100752457656, 12.65389680483651, 13.23710534059764, 13.75065902039293, 14.45509322878760, 14.77794275997982

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