Properties

Label 2-5225-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.97·2-s − 2.65·3-s + 1.91·4-s + 5.24·6-s − 1.06·7-s + 0.173·8-s + 4.04·9-s + 11-s − 5.07·12-s − 0.939·13-s + 2.10·14-s − 4.16·16-s − 3.65·17-s − 7.99·18-s − 19-s + 2.83·21-s − 1.97·22-s − 5.35·23-s − 0.461·24-s + 1.85·26-s − 2.77·27-s − 2.03·28-s − 4.46·29-s − 2.71·31-s + 7.89·32-s − 2.65·33-s + 7.23·34-s + ⋯
L(s)  = 1  − 1.39·2-s − 1.53·3-s + 0.956·4-s + 2.14·6-s − 0.403·7-s + 0.0614·8-s + 1.34·9-s + 0.301·11-s − 1.46·12-s − 0.260·13-s + 0.563·14-s − 1.04·16-s − 0.887·17-s − 1.88·18-s − 0.229·19-s + 0.617·21-s − 0.421·22-s − 1.11·23-s − 0.0941·24-s + 0.364·26-s − 0.533·27-s − 0.385·28-s − 0.829·29-s − 0.488·31-s + 1.39·32-s − 0.462·33-s + 1.24·34-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: \(0\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1567766788\)
\(L(\frac12)\) \(\approx\) \(0.1567766788\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.97T + 2T^{2} \)
3 \( 1 + 2.65T + 3T^{2} \)
7 \( 1 + 1.06T + 7T^{2} \)
13 \( 1 + 0.939T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
23 \( 1 + 5.35T + 23T^{2} \)
29 \( 1 + 4.46T + 29T^{2} \)
31 \( 1 + 2.71T + 31T^{2} \)
37 \( 1 + 0.932T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 - 4.30T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + 6.15T + 59T^{2} \)
61 \( 1 - 2.13T + 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 - 7.32T + 71T^{2} \)
73 \( 1 + 1.14T + 73T^{2} \)
79 \( 1 + 6.71T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 8.77T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.295765010281951916953585148904, −7.36703343656350192094675101875, −6.90149801242154099133516199619, −6.19161788821031328751312680566, −5.55452292890131446377310261464, −4.60464140939973623303533456949, −3.91315238176757996545086473735, −2.37569294345999147922085556059, −1.43284284743381462549774821195, −0.30162325571580934383144937732, 0.30162325571580934383144937732, 1.43284284743381462549774821195, 2.37569294345999147922085556059, 3.91315238176757996545086473735, 4.60464140939973623303533456949, 5.55452292890131446377310261464, 6.19161788821031328751312680566, 6.90149801242154099133516199619, 7.36703343656350192094675101875, 8.295765010281951916953585148904

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