Properties

Label 2-47190-1.1-c1-0-6
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 $0$

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 − 18-s − 19-s − 20-s − 4·21-s + 6·23-s − 24-s + 25-s − 26-s + 27-s − 4·28-s + 3·29-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 − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.872·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.557·29-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: \(0\)
Selberg data: \((2,\ 47190,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573522750\)
\(L(\frac12)\) \(\approx\) \(1.573522750\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.62457019797482, −14.11214989652025, −13.47742871780286, −13.05152080983498, −12.46548869246722, −12.17395166847822, −11.34683583812406, −10.90291234944797, −10.25738600203521, −9.813107331426677, −9.380256169528623, −8.808651357404452, −8.347323955887027, −7.855878061877253, −6.974309495369434, −6.839449456839207, −6.238707401062290, −5.509554279404629, −4.672156010263866, −3.943126904384749, −3.355188639175831, −2.823044330743638, −2.309808871313106, −1.139580196969833, −0.5479244431792825, 0.5479244431792825, 1.139580196969833, 2.309808871313106, 2.823044330743638, 3.355188639175831, 3.943126904384749, 4.672156010263866, 5.509554279404629, 6.238707401062290, 6.839449456839207, 6.974309495369434, 7.855878061877253, 8.347323955887027, 8.808651357404452, 9.380256169528623, 9.813107331426677, 10.25738600203521, 10.90291234944797, 11.34683583812406, 12.17395166847822, 12.46548869246722, 13.05152080983498, 13.47742871780286, 14.11214989652025, 14.62457019797482

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