Properties

Label 2-5225-1.1-c1-0-137
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  − 0.745·2-s + 2.05·3-s − 1.44·4-s − 1.52·6-s + 3.86·7-s + 2.56·8-s + 1.20·9-s + 11-s − 2.96·12-s + 2.04·13-s − 2.88·14-s + 0.975·16-s + 3.70·17-s − 0.898·18-s − 19-s + 7.93·21-s − 0.745·22-s − 2.09·23-s + 5.26·24-s − 1.52·26-s − 3.68·27-s − 5.58·28-s − 1.56·29-s + 5.38·31-s − 5.86·32-s + 2.05·33-s − 2.75·34-s + ⋯
L(s)  = 1  − 0.527·2-s + 1.18·3-s − 0.722·4-s − 0.623·6-s + 1.46·7-s + 0.907·8-s + 0.401·9-s + 0.301·11-s − 0.855·12-s + 0.566·13-s − 0.770·14-s + 0.243·16-s + 0.898·17-s − 0.211·18-s − 0.229·19-s + 1.73·21-s − 0.158·22-s − 0.436·23-s + 1.07·24-s − 0.298·26-s − 0.708·27-s − 1.05·28-s − 0.290·29-s + 0.967·31-s − 1.03·32-s + 0.356·33-s − 0.473·34-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

\(L(1)\) \(\approx\) \(2.546533920\)
\(L(\frac12)\) \(\approx\) \(2.546533920\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.745T + 2T^{2} \)
3 \( 1 - 2.05T + 3T^{2} \)
7 \( 1 - 3.86T + 7T^{2} \)
13 \( 1 - 2.04T + 13T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
23 \( 1 + 2.09T + 23T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 - 5.38T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 - 5.29T + 47T^{2} \)
53 \( 1 + 2.10T + 53T^{2} \)
59 \( 1 - 0.643T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 7.38T + 67T^{2} \)
71 \( 1 + 3.12T + 71T^{2} \)
73 \( 1 + 6.41T + 73T^{2} \)
79 \( 1 + 3.06T + 79T^{2} \)
83 \( 1 + 6.28T + 83T^{2} \)
89 \( 1 + 1.85T + 89T^{2} \)
97 \( 1 + 6.58T + 97T^{2} \)
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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

−8.372664108640047320595127167024, −7.85935682620359524418745607676, −7.22445564430629846598120910470, −5.94749225951646328777730720380, −5.20755969294771604580772259373, −4.28674120761649065753033242939, −3.84758076746054196312086651082, −2.73902300161125053050756811971, −1.76320167282717656724329630009, −0.972020248521000734715909997927, 0.972020248521000734715909997927, 1.76320167282717656724329630009, 2.73902300161125053050756811971, 3.84758076746054196312086651082, 4.28674120761649065753033242939, 5.20755969294771604580772259373, 5.94749225951646328777730720380, 7.22445564430629846598120910470, 7.85935682620359524418745607676, 8.372664108640047320595127167024

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