Properties

Label 2-5225-1.1-c1-0-194
Degree $2$
Conductor $5225$
Sign $-1$
Analytic cond. $41.7218$
Root an. cond. $6.45924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.759·2-s + 2.25·3-s − 1.42·4-s − 1.71·6-s − 4.95·7-s + 2.59·8-s + 2.09·9-s − 11-s − 3.21·12-s − 0.499·13-s + 3.75·14-s + 0.874·16-s + 4.34·17-s − 1.59·18-s − 19-s − 11.1·21-s + 0.759·22-s + 2.78·23-s + 5.86·24-s + 0.378·26-s − 2.03·27-s + 7.05·28-s + 8.84·29-s + 1.67·31-s − 5.86·32-s − 2.25·33-s − 3.30·34-s + ⋯
L(s)  = 1  − 0.536·2-s + 1.30·3-s − 0.711·4-s − 0.699·6-s − 1.87·7-s + 0.918·8-s + 0.699·9-s − 0.301·11-s − 0.928·12-s − 0.138·13-s + 1.00·14-s + 0.218·16-s + 1.05·17-s − 0.375·18-s − 0.229·19-s − 2.44·21-s + 0.161·22-s + 0.581·23-s + 1.19·24-s + 0.0743·26-s − 0.391·27-s + 1.33·28-s + 1.64·29-s + 0.300·31-s − 1.03·32-s − 0.393·33-s − 0.566·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(41.7218\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + 0.759T + 2T^{2} \)
3 \( 1 - 2.25T + 3T^{2} \)
7 \( 1 + 4.95T + 7T^{2} \)
13 \( 1 + 0.499T + 13T^{2} \)
17 \( 1 - 4.34T + 17T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 - 8.84T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 - 1.81T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 2.84T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + 7.25T + 89T^{2} \)
97 \( 1 + 0.0194T + 97T^{2} \)
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

−8.021915476900336593482756204631, −7.40938273848264079437339743626, −6.57170120847441914632015377838, −5.78464161799476837380414490755, −4.72937850190224331406413743471, −3.87556241984402427231238466473, −3.09205466247752282732067794925, −2.72171306558403445201561688595, −1.23436648291945867257609904186, 0, 1.23436648291945867257609904186, 2.72171306558403445201561688595, 3.09205466247752282732067794925, 3.87556241984402427231238466473, 4.72937850190224331406413743471, 5.78464161799476837380414490755, 6.57170120847441914632015377838, 7.40938273848264079437339743626, 8.021915476900336593482756204631

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