↑

Properties

Label 2-379050-1.1-c1-0-125
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 + 3·11-s − 12-s + 13-s − 14-s + 16-s + 18-s + 21-s + 3·22-s + 3·23-s − 24-s + 26-s − 27-s − 28-s + 8·29-s + 9·31-s + 32-s − 3·33-s + 36-s + 8·37-s − 39-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 + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.639·22-s + 0.625·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 1.61·31-s + 0.176·32-s − 0.522·33-s + 1/6·36-s + 1.31·37-s − 0.160·39-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

\(L(1)\) \(\approx\) \(5.411642904\)
\(L(\frac12)\) \(\approx\) \(5.411642904\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−12.42149671809689, −12.02636199612805, −11.67109400055285, −11.28640330588552, −10.79214915531316, −10.19801526402509, −9.960761391384688, −9.422940671229464, −8.771042945019947, −8.421982669341151, −7.843234551035408, −7.227242827389876, −6.709369088979042, −6.449023274823905, −6.079271596155181, −5.496678756621528, −4.892700544194566, −4.552544629097205, −4.033278045615687, −3.520733955162889, −2.917515739576753, −2.474926692052714, −1.693167940941971, −0.9742358504675478, −0.6667681441076929, 0.6667681441076929, 0.9742358504675478, 1.693167940941971, 2.474926692052714, 2.917515739576753, 3.520733955162889, 4.033278045615687, 4.552544629097205, 4.892700544194566, 5.496678756621528, 6.079271596155181, 6.449023274823905, 6.709369088979042, 7.227242827389876, 7.843234551035408, 8.421982669341151, 8.771042945019947, 9.422940671229464, 9.960761391384688, 10.19801526402509, 10.79214915531316, 11.28640330588552, 11.67109400055285, 12.02636199612805, 12.42149671809689

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