Properties

Label 2-3525-1.1-c1-0-23
Degree $2$
Conductor $3525$
Sign $1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
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 − 4·7-s − 3·8-s + 9-s − 12-s − 6·13-s − 4·14-s − 16-s + 6·17-s + 18-s + 2·19-s − 4·21-s − 4·23-s − 3·24-s − 6·26-s + 27-s + 4·28-s + 8·29-s + 6·31-s + 5·32-s + 6·34-s − 36-s + 6·37-s + 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s − 0.834·23-s − 0.612·24-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.48·29-s + 1.07·31-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.986·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.852212391\)
\(L(\frac12)\) \(\approx\) \(1.852212391\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
47 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−8.543359650662236098417331516923, −7.86786008362572605457503887932, −6.99294957245247344706144578433, −6.26711155117347263403320630315, −5.46249218413326480091048866367, −4.68802659153404688563028812078, −3.83440269089398061389622899594, −3.06046767146419013698609448578, −2.57098947113755432400667040759, −0.67892565669524491181453663738, 0.67892565669524491181453663738, 2.57098947113755432400667040759, 3.06046767146419013698609448578, 3.83440269089398061389622899594, 4.68802659153404688563028812078, 5.46249218413326480091048866367, 6.26711155117347263403320630315, 6.99294957245247344706144578433, 7.86786008362572605457503887932, 8.543359650662236098417331516923

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