Properties

Label 2-6025-1.1-c1-0-349
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  + 2.60·2-s + 3.03·3-s + 4.76·4-s + 7.87·6-s + 1.49·7-s + 7.18·8-s + 6.18·9-s − 4.55·11-s + 14.4·12-s + 6.26·13-s + 3.89·14-s + 9.15·16-s − 5.97·17-s + 16.0·18-s − 3.57·19-s + 4.54·21-s − 11.8·22-s + 6.76·23-s + 21.7·24-s + 16.2·26-s + 9.64·27-s + 7.13·28-s − 3.27·29-s − 4.86·31-s + 9.44·32-s − 13.7·33-s − 15.5·34-s + ⋯
L(s)  = 1  + 1.83·2-s + 1.74·3-s + 2.38·4-s + 3.21·6-s + 0.566·7-s + 2.53·8-s + 2.06·9-s − 1.37·11-s + 4.16·12-s + 1.73·13-s + 1.04·14-s + 2.28·16-s − 1.44·17-s + 3.78·18-s − 0.821·19-s + 0.991·21-s − 2.52·22-s + 1.41·23-s + 4.44·24-s + 3.19·26-s + 1.85·27-s + 1.34·28-s − 0.608·29-s − 0.873·31-s + 1.66·32-s − 2.40·33-s − 2.66·34-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

\(L(1)\) \(\approx\) \(12.92244961\)
\(L(\frac12)\) \(\approx\) \(12.92244961\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 2T^{2} \)
3 \( 1 - 3.03T + 3T^{2} \)
7 \( 1 - 1.49T + 7T^{2} \)
11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 - 6.26T + 13T^{2} \)
17 \( 1 + 5.97T + 17T^{2} \)
19 \( 1 + 3.57T + 19T^{2} \)
23 \( 1 - 6.76T + 23T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + 4.86T + 31T^{2} \)
37 \( 1 + 9.31T + 37T^{2} \)
41 \( 1 - 1.97T + 41T^{2} \)
43 \( 1 + 5.98T + 43T^{2} \)
47 \( 1 - 8.42T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 8.77T + 59T^{2} \)
61 \( 1 - 9.19T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 - 1.35T + 73T^{2} \)
79 \( 1 - 0.518T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 6.25T + 89T^{2} \)
97 \( 1 - 7.07T + 97T^{2} \)
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

−8.013521469893032147891924733787, −7.22466052636444802042329047294, −6.69673953635084448443005425260, −5.75261361767763012406929614862, −4.92855101609846072811227236951, −4.32543325963446080418217822153, −3.56055891261628770762420521225, −3.07451327450340311135727556322, −2.18807540666911769617487427719, −1.70466223694656692743793192469, 1.70466223694656692743793192469, 2.18807540666911769617487427719, 3.07451327450340311135727556322, 3.56055891261628770762420521225, 4.32543325963446080418217822153, 4.92855101609846072811227236951, 5.75261361767763012406929614862, 6.69673953635084448443005425260, 7.22466052636444802042329047294, 8.013521469893032147891924733787

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