Properties

Label 2-4025-1.1-c1-0-113
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  + 2.52·2-s − 1.80·3-s + 4.39·4-s − 4.56·6-s + 7-s + 6.05·8-s + 0.258·9-s + 3.95·11-s − 7.93·12-s − 0.378·13-s + 2.52·14-s + 6.53·16-s + 3.06·17-s + 0.654·18-s + 3.83·19-s − 1.80·21-s + 10.0·22-s − 23-s − 10.9·24-s − 0.958·26-s + 4.94·27-s + 4.39·28-s − 0.0915·29-s − 8.53·31-s + 4.39·32-s − 7.14·33-s + 7.75·34-s + ⋯
L(s)  = 1  + 1.78·2-s − 1.04·3-s + 2.19·4-s − 1.86·6-s + 0.377·7-s + 2.14·8-s + 0.0862·9-s + 1.19·11-s − 2.29·12-s − 0.105·13-s + 0.675·14-s + 1.63·16-s + 0.743·17-s + 0.154·18-s + 0.879·19-s − 0.393·21-s + 2.13·22-s − 0.208·23-s − 2.23·24-s − 0.187·26-s + 0.952·27-s + 0.830·28-s − 0.0169·29-s − 1.53·31-s + 0.777·32-s − 1.24·33-s + 1.32·34-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

\(L(1)\) \(\approx\) \(4.740006969\)
\(L(\frac12)\) \(\approx\) \(4.740006969\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.52T + 2T^{2} \)
3 \( 1 + 1.80T + 3T^{2} \)
11 \( 1 - 3.95T + 11T^{2} \)
13 \( 1 + 0.378T + 13T^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
29 \( 1 + 0.0915T + 29T^{2} \)
31 \( 1 + 8.53T + 31T^{2} \)
37 \( 1 - 9.98T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 4.07T + 43T^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 - 5.74T + 59T^{2} \)
61 \( 1 - 2.08T + 61T^{2} \)
67 \( 1 - 1.88T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 2.71T + 79T^{2} \)
83 \( 1 + 1.49T + 83T^{2} \)
89 \( 1 - 2.16T + 89T^{2} \)
97 \( 1 - 8.04T + 97T^{2} \)
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

−8.177329760844006325723743243081, −7.22746408680892065234224408842, −6.67028267658882527532655110507, −5.94097921983145758952559813125, −5.39646960457792813351675351511, −4.87621525500550937513068304449, −3.93467415366724349615944950846, −3.36834853212648224527119983192, −2.19336282765768750153327382169, −1.06679172347757760854599377959, 1.06679172347757760854599377959, 2.19336282765768750153327382169, 3.36834853212648224527119983192, 3.93467415366724349615944950846, 4.87621525500550937513068304449, 5.39646960457792813351675351511, 5.94097921983145758952559813125, 6.67028267658882527532655110507, 7.22746408680892065234224408842, 8.177329760844006325723743243081

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