Properties

Label 2-3850-1.1-c1-0-25
Degree $2$
Conductor $3850$
Sign $1$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
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 + 4-s − 7-s + 8-s − 3·9-s − 11-s − 3·13-s − 14-s + 16-s + 2·17-s − 3·18-s + 5·19-s − 22-s + 7·23-s − 3·26-s − 28-s + 3·29-s − 3·31-s + 32-s + 2·34-s − 3·36-s + 8·37-s + 5·38-s − 11·43-s − 44-s + 7·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.14·19-s − 0.213·22-s + 1.45·23-s − 0.588·26-s − 0.188·28-s + 0.557·29-s − 0.538·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.31·37-s + 0.811·38-s − 1.67·43-s − 0.150·44-s + 1.03·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.580355754\)
\(L(\frac12)\) \(\approx\) \(2.580355754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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

−8.352054853371123821033284853939, −7.65558100473630831122600941631, −6.93027941342444787375688904823, −6.19869388198090898681557576358, −5.16057128352108511744525826145, −5.11183024607625390725057160754, −3.71651330528317709288293448594, −3.02510092100263003661086566599, −2.36294746398274522663361775067, −0.823263731924689320087869435652, 0.823263731924689320087869435652, 2.36294746398274522663361775067, 3.02510092100263003661086566599, 3.71651330528317709288293448594, 5.11183024607625390725057160754, 5.16057128352108511744525826145, 6.19869388198090898681557576358, 6.93027941342444787375688904823, 7.65558100473630831122600941631, 8.352054853371123821033284853939

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