↑

Properties

Label 2-379050-1.1-c1-0-106
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 5·11-s + 12-s + 4·13-s − 14-s + 16-s + 5·17-s − 18-s + 21-s + 5·22-s + 2·23-s − 24-s − 4·26-s + 27-s + 28-s + 6·29-s − 32-s − 5·33-s − 5·34-s + 36-s + 2·37-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 − 1.50·11-s + 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s + 0.417·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.176·32-s − 0.870·33-s − 0.857·34-s + 1/6·36-s + 0.328·37-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

\(L(1)\) \(\approx\) \(3.463160883\)
\(L(\frac12)\) \(\approx\) \(3.463160883\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.56173309561743, −12.11018598674655, −11.31373747243981, −11.05776673480608, −10.63797082253731, −10.13843247896357, −9.886775124501616, −9.158454060776640, −8.867174801196321, −8.313514666170213, −7.958754656176568, −7.600365477679906, −7.260512503745212, −6.518138573842599, −5.943873379634704, −5.664886193919399, −4.950734521204993, −4.539563817873165, −3.798109717653443, −3.268885430841137, −2.783679760180841, −2.356743515073771, −1.663814981848129, −0.9882481446400619, −0.6103659587537479, 0.6103659587537479, 0.9882481446400619, 1.663814981848129, 2.356743515073771, 2.783679760180841, 3.268885430841137, 3.798109717653443, 4.539563817873165, 4.950734521204993, 5.664886193919399, 5.943873379634704, 6.518138573842599, 7.260512503745212, 7.600365477679906, 7.958754656176568, 8.313514666170213, 8.867174801196321, 9.158454060776640, 9.886775124501616, 10.13843247896357, 10.63797082253731, 11.05776673480608, 11.31373747243981, 12.11018598674655, 12.56173309561743

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