Properties

Label 2-47190-1.1-c1-0-34
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 + 4·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 4·14-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 24-s + 25-s − 26-s − 27-s + 4·28-s − 2·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 1.06·14-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 − 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.182·30-s + 1.43·31-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.04443216373263, −14.31453114506954, −14.00391446495491, −13.24577370215019, −12.58006947023608, −12.11354027197358, −11.59993738639396, −11.18807572295285, −10.66735677077686, −10.48869907110920, −9.596036311254388, −8.955994691844238, −8.527732904581903, −7.947424766965453, −7.616287884924208, −6.918495880683056, −6.330407230332641, −5.785014410093779, −5.020056194799134, −4.492197621808360, −4.066742610857728, −3.070805976575230, −2.236668947618536, −1.615587497654908, −0.9324994099503916, 0, 0.9324994099503916, 1.615587497654908, 2.236668947618536, 3.070805976575230, 4.066742610857728, 4.492197621808360, 5.020056194799134, 5.785014410093779, 6.330407230332641, 6.918495880683056, 7.616287884924208, 7.947424766965453, 8.527732904581903, 8.955994691844238, 9.596036311254388, 10.48869907110920, 10.66735677077686, 11.18807572295285, 11.59993738639396, 12.11354027197358, 12.58006947023608, 13.24577370215019, 14.00391446495491, 14.31453114506954, 15.04443216373263

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