Properties

Label 2-47190-1.1-c1-0-45
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 + 8·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s − 2·29-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 − 1.51·7-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 + 1.66·23-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 + ⋯

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: Trivial
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 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.85987072730676, −14.52930546074750, −13.59676701652196, −13.18175786457411, −13.02986694632963, −12.41127551655673, −11.99578774037004, −11.14462611291151, −10.84564499988848, −10.15212092620135, −9.633083163702873, −9.064378407255781, −8.604420354838946, −7.934559284457693, −7.105837974589088, −6.987217664062354, −6.319913803919769, −5.643149477055904, −5.116480594771980, −4.198919997399951, −3.815482466125074, −3.191564440407635, −2.823875090125015, −1.994569303907133, −1.028349566406169, 0, 1.028349566406169, 1.994569303907133, 2.823875090125015, 3.191564440407635, 3.815482466125074, 4.198919997399951, 5.116480594771980, 5.643149477055904, 6.319913803919769, 6.987217664062354, 7.105837974589088, 7.934559284457693, 8.604420354838946, 9.064378407255781, 9.633083163702873, 10.15212092620135, 10.84564499988848, 11.14462611291151, 11.99578774037004, 12.41127551655673, 13.02986694632963, 13.18175786457411, 13.59676701652196, 14.52930546074750, 14.85987072730676

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