Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 2.37·3-s + 1.01·4-s − 4.11·6-s + 2.01·7-s + 1.70·8-s + 2.62·9-s − 3.39·11-s + 2.41·12-s − 5.63·13-s − 3.49·14-s − 4.99·16-s − 0.866·17-s − 4.55·18-s + 2.46·19-s + 4.76·21-s + 5.89·22-s + 6.37·23-s + 4.05·24-s + 9.79·26-s − 0.892·27-s + 2.04·28-s − 4.52·29-s − 3.51·31-s + 5.26·32-s − 8.05·33-s + 1.50·34-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.36·3-s + 0.508·4-s − 1.68·6-s + 0.759·7-s + 0.603·8-s + 0.874·9-s − 1.02·11-s + 0.695·12-s − 1.56·13-s − 0.933·14-s − 1.24·16-s − 0.210·17-s − 1.07·18-s + 0.565·19-s + 1.04·21-s + 1.25·22-s + 1.33·23-s + 0.826·24-s + 1.91·26-s − 0.171·27-s + 0.386·28-s − 0.839·29-s − 0.631·31-s + 0.931·32-s − 1.40·33-s + 0.258·34-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 2T^{2} \)
3 \( 1 - 2.37T + 3T^{2} \)
7 \( 1 - 2.01T + 7T^{2} \)
11 \( 1 + 3.39T + 11T^{2} \)
13 \( 1 + 5.63T + 13T^{2} \)
17 \( 1 + 0.866T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 - 6.37T + 23T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 - 5.19T + 37T^{2} \)
41 \( 1 - 1.35T + 41T^{2} \)
43 \( 1 + 8.49T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 - 6.03T + 59T^{2} \)
61 \( 1 - 4.45T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 3.01T + 71T^{2} \)
73 \( 1 - 0.255T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + 0.273T + 97T^{2} \)
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

−7.85597771402575344655649581680, −7.47094555812268275016669332737, −6.90388147386676771306516765865, −5.31770471834814322012216312978, −4.90609689019943834485586877600, −3.91690973256586203223628602336, −2.77917796061170350019401807369, −2.31185049944708379447400409459, −1.38475331285569718575612177361, 0, 1.38475331285569718575612177361, 2.31185049944708379447400409459, 2.77917796061170350019401807369, 3.91690973256586203223628602336, 4.90609689019943834485586877600, 5.31770471834814322012216312978, 6.90388147386676771306516765865, 7.47094555812268275016669332737, 7.85597771402575344655649581680

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