Properties

Label 2-6025-1.1-c1-0-3
Degree $2$
Conductor $6025$
Sign $1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.38·2-s − 1.64·3-s − 0.0828·4-s + 2.27·6-s − 2.06·7-s + 2.88·8-s − 0.288·9-s − 1.34·11-s + 0.136·12-s + 0.417·13-s + 2.85·14-s − 3.82·16-s − 7.21·17-s + 0.399·18-s − 4.92·19-s + 3.39·21-s + 1.85·22-s + 7.81·23-s − 4.74·24-s − 0.578·26-s + 5.41·27-s + 0.170·28-s − 5.56·29-s − 6.74·31-s − 0.468·32-s + 2.20·33-s + 9.99·34-s + ⋯
L(s)  = 1  − 0.979·2-s − 0.950·3-s − 0.0414·4-s + 0.930·6-s − 0.779·7-s + 1.01·8-s − 0.0962·9-s − 0.404·11-s + 0.0393·12-s + 0.115·13-s + 0.763·14-s − 0.956·16-s − 1.75·17-s + 0.0942·18-s − 1.12·19-s + 0.740·21-s + 0.395·22-s + 1.62·23-s − 0.969·24-s − 0.113·26-s + 1.04·27-s + 0.0322·28-s − 1.03·29-s − 1.21·31-s − 0.0827·32-s + 0.384·33-s + 1.71·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01344457729\)
\(L(\frac12)\) \(\approx\) \(0.01344457729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 + T \)
good2 \( 1 + 1.38T + 2T^{2} \)
3 \( 1 + 1.64T + 3T^{2} \)
7 \( 1 + 2.06T + 7T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 - 0.417T + 13T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 + 4.92T + 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 + 5.56T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 8.08T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 8.61T + 43T^{2} \)
47 \( 1 + 9.63T + 47T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 - 4.14T + 61T^{2} \)
67 \( 1 + 6.95T + 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + 3.70T + 79T^{2} \)
83 \( 1 - 1.93T + 83T^{2} \)
89 \( 1 + 4.50T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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

−8.420382607084459822848553172277, −7.21011581339820345247167787880, −6.78955962968880734681435796309, −6.12159655204364137572510181079, −5.16875322697357583528874984975, −4.66354055622118080231044802784, −3.68666447757731128943998947165, −2.58082422408542266968985771596, −1.52117407751635688201705691378, −0.07499689058893673073988574290, 0.07499689058893673073988574290, 1.52117407751635688201705691378, 2.58082422408542266968985771596, 3.68666447757731128943998947165, 4.66354055622118080231044802784, 5.16875322697357583528874984975, 6.12159655204364137572510181079, 6.78955962968880734681435796309, 7.21011581339820345247167787880, 8.420382607084459822848553172277

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