Properties

Label 2-47190-1.1-c1-0-54
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 − 8-s + 9-s − 10-s − 12-s + 13-s − 15-s + 16-s + 2·17-s − 18-s + 4·19-s + 20-s + 2·23-s + 24-s + 25-s − 26-s − 27-s + 6·29-s + 30-s − 32-s − 2·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.328·37-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−15.04614351958739, −14.31355194905168, −13.68163147694453, −13.47534851994519, −12.56324768827032, −12.18136462254052, −11.76626244351340, −11.11981270039521, −10.57903358160374, −10.27119017857736, −9.590363061729302, −9.213727943420524, −8.619014705125723, −7.909355191568750, −7.512672341743437, −6.768478824713615, −6.400691589258106, −5.714755934336287, −5.248616025635366, −4.611117705544412, −3.775978940314519, −3.025748346856269, −2.455432522417944, −1.396015163492792, −1.083129683694531, 0, 1.083129683694531, 1.396015163492792, 2.455432522417944, 3.025748346856269, 3.775978940314519, 4.611117705544412, 5.248616025635366, 5.714755934336287, 6.400691589258106, 6.768478824713615, 7.512672341743437, 7.909355191568750, 8.619014705125723, 9.213727943420524, 9.590363061729302, 10.27119017857736, 10.57903358160374, 11.11981270039521, 11.76626244351340, 12.18136462254052, 12.56324768827032, 13.47534851994519, 13.68163147694453, 14.31355194905168, 15.04614351958739

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