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Properties

Label 2-379050-1.1-c1-0-1
Degree $2$
Conductor $379050$
Sign $1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
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 − 6-s + 7-s − 8-s + 9-s − 3·11-s + 12-s − 14-s + 16-s − 5·17-s − 18-s + 21-s + 3·22-s − 6·23-s − 24-s + 27-s + 28-s + 4·29-s − 8·31-s − 32-s − 3·33-s + 5·34-s + 36-s + 4·37-s + 7·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s + 0.192·27-s + 0.188·28-s + 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.522·33-s + 0.857·34-s + 1/6·36-s + 0.657·37-s + 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4512561947\)
\(L(\frac12)\) \(\approx\) \(0.4512561947\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−12.36426733921131, −12.10943899581665, −11.33796504868066, −11.05319499783488, −10.73719016168139, −10.08110338876590, −9.744275544669887, −9.361537880238330, −8.700434836306948, −8.420733997779529, −7.988076185368106, −7.632461530473812, −6.999607036921686, −6.719948447111129, −5.989751659423896, −5.594756813415006, −4.981495943329785, −4.357655564662689, −4.010546881003651, −3.290859552180103, −2.636212210131334, −2.329968201731758, −1.746168623205236, −1.180220785482423, −0.1827520950237345, 0.1827520950237345, 1.180220785482423, 1.746168623205236, 2.329968201731758, 2.636212210131334, 3.290859552180103, 4.010546881003651, 4.357655564662689, 4.981495943329785, 5.594756813415006, 5.989751659423896, 6.719948447111129, 6.999607036921686, 7.632461530473812, 7.988076185368106, 8.420733997779529, 8.700434836306948, 9.361537880238330, 9.744275544669887, 10.08110338876590, 10.73719016168139, 11.05319499783488, 11.33796504868066, 12.10943899581665, 12.36426733921131

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