Properties

Label 2-3850-1.1-c1-0-82
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 3·9-s − 11-s − 2·13-s + 14-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 22-s − 4·23-s − 2·26-s + 28-s − 2·29-s + 8·31-s + 32-s − 6·34-s − 3·36-s + 10·37-s + 4·38-s − 6·41-s − 12·43-s − 44-s − 4·46-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 − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.213·22-s − 0.834·23-s − 0.392·26-s + 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.82·43-s − 0.150·44-s − 0.589·46-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: $\chi_{3850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−8.091572992705813150503503841520, −7.36724638720596046836615184349, −6.41444614335844704712814652637, −5.92691045159899202485708703134, −4.87357292365988070618620622182, −4.59161092301521288709471744001, −3.32451610962817455478500208221, −2.66903939783330043504079877356, −1.71052969422956014629545325941, 0, 1.71052969422956014629545325941, 2.66903939783330043504079877356, 3.32451610962817455478500208221, 4.59161092301521288709471744001, 4.87357292365988070618620622182, 5.92691045159899202485708703134, 6.41444614335844704712814652637, 7.36724638720596046836615184349, 8.091572992705813150503503841520

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