Properties

Label 2-4025-1.1-c1-0-56
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 2.23·3-s − 1.85·4-s + 0.854·6-s + 7-s + 1.47·8-s + 2.00·9-s + 6·11-s + 4.14·12-s − 3·13-s − 0.381·14-s + 3.14·16-s + 1.23·17-s − 0.763·18-s + 8.47·19-s − 2.23·21-s − 2.29·22-s − 23-s − 3.29·24-s + 1.14·26-s + 2.23·27-s − 1.85·28-s + 7.47·29-s − 5.47·31-s − 4.14·32-s − 13.4·33-s − 0.472·34-s + ⋯
L(s)  = 1  − 0.270·2-s − 1.29·3-s − 0.927·4-s + 0.348·6-s + 0.377·7-s + 0.520·8-s + 0.666·9-s + 1.80·11-s + 1.19·12-s − 0.832·13-s − 0.102·14-s + 0.786·16-s + 0.299·17-s − 0.180·18-s + 1.94·19-s − 0.487·21-s − 0.488·22-s − 0.208·23-s − 0.671·24-s + 0.224·26-s + 0.430·27-s − 0.350·28-s + 1.38·29-s − 0.982·31-s − 0.732·32-s − 2.33·33-s − 0.0809·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9794527235\)
\(L(\frac12)\) \(\approx\) \(0.9794527235\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 - 8.47T + 19T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 + 3.70T + 43T^{2} \)
47 \( 1 - 4.23T + 47T^{2} \)
53 \( 1 - 8.76T + 53T^{2} \)
59 \( 1 + 6.47T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 + 3.23T + 67T^{2} \)
71 \( 1 + 4.23T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 3.70T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 - 0.291T + 97T^{2} \)
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

−8.590407744167468467415575538495, −7.55128996321019447817606720708, −7.06880638163747721269805870845, −6.00698747081970831777559219949, −5.53841318456108248607639875171, −4.67966248915388153685711943317, −4.20167464567717119411473056639, −3.09143508365505756202329452990, −1.38270441315252398526850379063, −0.73740859502333881255175797230, 0.73740859502333881255175797230, 1.38270441315252398526850379063, 3.09143508365505756202329452990, 4.20167464567717119411473056639, 4.67966248915388153685711943317, 5.53841318456108248607639875171, 6.00698747081970831777559219949, 7.06880638163747721269805870845, 7.55128996321019447817606720708, 8.590407744167468467415575538495

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