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Properties

Label 2-379050-1.1-c1-0-103
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 + 2·11-s + 12-s + 4·13-s + 14-s + 16-s − 3·17-s − 18-s − 21-s − 2·22-s − 24-s − 4·26-s + 27-s − 28-s + 6·29-s + 8·31-s − 32-s + 2·33-s + 3·34-s + 36-s + 11·37-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 + 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.218·21-s − 0.426·22-s − 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + 1.80·37-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

\(L(1)\) \(\approx\) \(3.441935469\)
\(L(\frac12)\) \(\approx\) \(3.441935469\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.29489416253988, −12.09491678013266, −11.44949264780671, −11.08199509549462, −10.62887547414772, −10.12852568373665, −9.728039623964696, −9.149119807583346, −8.935232874757386, −8.477633542854669, −7.965644788853257, −7.602353721291029, −7.000248971322932, −6.411829004924410, −6.205512012204113, −5.808369402122128, −4.768598287306993, −4.442917646151283, −3.883037355237475, −3.357519236061146, −2.643168226237730, −2.481893068620475, −1.590003658243772, −1.019330112353741, −0.6044461903954026, 0.6044461903954026, 1.019330112353741, 1.590003658243772, 2.481893068620475, 2.643168226237730, 3.357519236061146, 3.883037355237475, 4.442917646151283, 4.768598287306993, 5.808369402122128, 6.205512012204113, 6.411829004924410, 7.000248971322932, 7.602353721291029, 7.965644788853257, 8.477633542854669, 8.935232874757386, 9.149119807583346, 9.728039623964696, 10.12852568373665, 10.62887547414772, 11.08199509549462, 11.44949264780671, 12.09491678013266, 12.29489416253988

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